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Section 1
Rules for Exponents
Fill in the blank.Expressions such as $x^{4}, 10^{3},$ and $(5 t)^{2}$ are called _____ expressions.
Fill in the blank.Match each expression below with the proper description on the next page.$$\frac{a^{8}}{a^{2}} \quad\left(a^{4} b^{2}\right)^{5} \quad\left(\frac{a^{6}}{a}\right)^{3} \quad\left(a^{8}\right)^{4} \quad a^{5} \cdot a^{3}$$A. Product of exponential expressions with the same baseB. Quotient of exponential expressions with the same baseC. Power of an exponential expressionD. Power of a productE. Power of a quotient
Fill in the blank.a. $(3 x)^{4}=$b. $(-5 y)(-5 y)(-5 y)=$
Fill in the blank.A. $x=x$B. $x^{m} x^{n}=$C. $(x y)^{n}=$D. $\left(a^{b}\right)^{c}=$E. $\frac{x^{m}}{x^{n}}=$F. $\left(\frac{a}{b}\right)^{n}=$
Fill in the blank.To simplify each expression, determine whether you add, subtract, multiply, or divide the exponents.A. $\frac{x^{8}}{x^{2}}$B. $b^{6} \cdot b^{9}$C. $\left(n^{8}\right)^{4}$D. $\left(a^{4} b^{2}\right)^{5}$
Fill in the blank.A. To simplify $\left(2 y^{3} z^{2}\right)^{4},$ what factors within the parentheses must be raised to the fourth power?B. To simplify $\left(\frac{y^{3}}{z^{2}}\right)^{4},$ what two expressions must be raised to the fourth power?
Simplify each expression, if possible.A. $x^{2}+x^{2}$B. $x^{2} \cdot x^{2}$C. $x^{2}+x$D. $x^{2} \cdot x$
Simplify each expression, if possible.A. $x^{3}-x^{2}$B. $\frac{x^{3}}{x^{2}}$C. $4^{2} \cdot 2^{4}$D. $\frac{x^{3}}{y^{2}}$
Complete each solution to simplify each expression.$$\left(x^{4} x^{2}\right)^{3}=(\quad)^{3}=x$$
Complete each solution to simplify each expression.$$\frac{a^{3} a^{4}}{a^{2}}=\frac{ }{a^{2}}=a^{-2}=a$$
Fill in the blanks.A. We read $9^{4}$ as "nine to the fourth _____."B. We read $\left(a^{2} b^{6}\right)\left(a^{4} b^{5}\right)$ as "the _____ of $a^{2} b^{6}$ times the ____ of $a^{4} b^{5}$C. We read $\left(3^{6}\right)^{9}$ as " 3 to the $\quad$ power, raised to the ____ power, raised to the _____ power:
Fill in the blanks.a. We read $n^{2} n^{3} n$ as "n _____ times $n$ _____M times $n$.b. We read $\frac{x^{7}}{x^{5}}$ as " $x$ to the seventh power ____ by $x$ to the _____ power:c. We read $(b+5)^{6}(b+5)^{8}$ as "the _____ of $b+5$ _____ to the sixth power, times the _____ of $b+5$ raised to the _____ power."
Identify the base and the exponent in each expression.A. $4^{3}$B. $-4^{3}$C. $(-4)^{3}$
Identify the base and the exponent in each expression.A. $x^{5}$B. $-x^{5}$C. $(-x)^{5}$
Identify the base and the exponent in each expression.A. $(-3 x)^{2}$B. $-3 x^{2}$C. $-(-3 x)^{2}$
Identify the base and the exponent in each expression.A. $-\frac{1}{3} x^{6}$B. $\left(-\frac{1}{3} x\right)^{6}$C. $-\left(-\frac{1}{3} x\right)^{6}$
Identify the base and the exponent in each expression.A. $9 m^{12}$B. $(9 m)^{12}$C. $-9 m^{12}$
Identify the base and the exponent in each expression.A. $(y+9)^{4}$B. $y+9^{4}$C. $8(y+9)^{4}$
Write each expression in an equivalent form using an exponent.$$4 t \cdot 4 t \cdot 4 t \cdot 4 t$$
Write each expression in an equivalent form using an exponent.$$-5 u(-5 u)(-5 u)(-5 u)(-5 u)$$
Write each expression in an equivalent form using an exponent.$$-4 \cdot t \cdot t \cdot t \cdot t \cdot t$$
Write each expression in an equivalent form using an exponent.$$-5 \cdot u \cdot u \cdot u$$
Write each expression in an equivalent form using an exponent.$$\frac{t}{2} \cdot \frac{t}{2} \cdot \frac{t}{2}$$
Write each expression in an equivalent form using an exponent.$$\frac{x}{c} \cdot \frac{x}{c} \cdot \frac{x}{c} \cdot \frac{x}{c}$$
Write each expression in an equivalent form using an exponent.$$(x-y)(x-y)$$
In Exercises 25 and $26,$ determine the time necessary for $\$ 1000$ to double if it is invested at interest rate $r$ compounded (a) annually,(b) monthly,(c) daily, and (d) continuously.$(m+4)(m+4)$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$5^{3} \cdot 5^{4}$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$3^{4} \cdot 3^{6}$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$b b^{2} b^{3}$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$a a^{3} a^{5}$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$(y-2)^{5}(y-2)^{2}$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$(t+1)^{5}(t+1)^{3}$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$\left(a^{2} b^{3}\right)\left(a^{3} b^{3}\right)$$
Use the product rule for exponents to simplify each expression. Write the results using exponents.$$\left(u^{3} v^{5}\right)\left(u^{4} v^{5}\right)$$
Find an expression that represents the area or volume of each figure. Recall that the formula for the volume of a rectangular solid is $V=$ length $\cdot$ width $\cdot$ height.(FIGURE NOT COPY)
Find an expression that represents the area or volume of each figure. Recall that the formula for the volume of a rectangular solid is $V=$ length $\cdot$ width $\cdot$ height.
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{8^{12}}{8^{4}}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{10^{4}}{10^{2}}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{x^{15}}{x^{3}}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{y^{6}}{y^{3}}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{(3.7 p)^{7}}{(3.7 p)^{2}}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{(0.25 y)^{9}}{(0.25 y)^{3}}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{c^{3} d^{7}}{c d}$$
Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$\frac{r^{8} s^{9}}{r s}$$
Use the product and quotient rules for exponents to simplify each expression.$$\frac{y^{3} y^{4}}{y y^{2}}$$
Use the product and quotient rules for exponents to simplify each expression.$$\frac{b^{4} b^{5}}{b^{2} b^{3}}$$
Use the product and quotient rules for exponents to simplify each expression.$$\frac{a^{2} a^{3} a^{4}}{a^{8}}$$
Use the product and quotient rules for exponents to simplify each expression.$$\frac{h^{3} h^{6} h}{h^{9}}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left(3^{2}\right)^{4}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left(4^{3}\right)^{3}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left[(-4.3)^{3}\right]^{8}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left[(-1.7)^{9}\right]^{8}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left(m^{50}\right)^{10}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left(n^{25}\right)^{4}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left(y^{5}\right)^{3}$$
Use the power rule for exponents to simplify each expression. Write the results using exponents.$$\left(b^{3}\right)^{6}$$
Use the product and power rules for exponents to simplify each expression.$$\left(x^{2} x^{3}\right)^{5}$$
Use the product and power rules for exponents to simplify each expression.$$\left(y^{3} y^{4}\right)^{4}$$
Use the product and power rules for exponents to simplify each expression.$$\left(p^{2} p^{3}\right)^{5}$$
Use the product and power rules for exponents to simplify each expression.$$\left(r^{3} r^{4}\right)^{2}$$
Use the product and power rules for exponents to simplify each expression.$$\left(t^{3}\right)^{4}\left(t^{2}\right)^{3}$$
Use the product and power rules for exponents to simplify each expression.$$\left(b^{2}\right)^{5}\left(b^{3}\right)^{2}$$
Use the product and power rules for exponents to simplify each expression.$$\left(u^{4}\right)^{2}\left(u^{3}\right)^{2}$$
Use the product and power rules for exponents to simplify each expression.$$\left(v^{5}\right)^{2}\left(v^{3}\right)^{4}$$
Use the power of a product rule for exponents to simplify each expression.$$(6 a)^{2}$$
Use the power of a product rule for exponents to simplify each expression.$$(3 b)^{3}$$
Use the power of a product rule for exponents to simplify each expression.$$(5 y)^{4}$$
Use the power of a product rule for exponents to simplify each expression.$$(4 t)^{4}$$
Use the power of a product rule for exponents to simplify each expression. See Example $9 .$$\left(-2 r^{2} s^{3}\right)^{3}$
Use the power of a product rule for exponents to simplify each expression.$$\left(-2 x^{2} y^{4}\right)^{5}$$
Use the power of a product rule for exponents to simplify each expression.$$\left(-\frac{1}{3} y^{2} z^{4}\right)^{5}$$
Use the power of a product rule for exponents to simplify each expression.$$\left(-\frac{1}{4} t^{3} u^{8}\right)^{2}$$
Use rules for exponents to simplify each expression.$$\frac{\left(a b^{2}\right)^{3}}{a^{2} b^{2}}$$
Use rules for exponents to simplify each expression.$$\frac{\left(m^{3} n^{4}\right)^{3}}{m^{3} n^{6}}$$
Use rules for exponents to simplify each expression.$$\frac{\left(r^{4} s^{3}\right)^{4}}{r^{3} s^{9}}$$
Use rules for exponents to simplify each expression.$$\frac{\left(x^{2} y^{5}\right)^{5}}{x^{6} y^{2}}$$
Use rules for exponents to simplify each expression.$$\frac{(6 k)^{7}}{(6 k)^{4}}$$
Use rules for exponents to simplify each expression.$$\frac{(-3 a)^{12}}{(-3 a)^{10}}$$
Use rules for exponents to simplify each expression.$$\frac{(3 q)^{5}}{(3 q)^{3}}$$
Use rules for exponents to simplify each expression.$$\frac{(a b)^{8}}{(a b)^{4}}$$
Use the power of a quotient rule for exponents to simplify each expression.$$\left(\frac{a}{b}\right)^{3}$$
Use the power of a quotient rule for exponents to simplify each expression.$$\left(\frac{r}{s}\right)^{4}$$
Use the power of a quotient rule for exponents to simplify each expression.$$\left(\frac{8 a^{2}}{11 b^{5}}\right)^{2}$$
Use the power of a quotient rule for exponents to simplify each expression.$$\left(\frac{7 g^{4}}{6 h^{3}}\right)^{2}$$
Simplify each expression, if possible.$$\left(\frac{x^{2}}{y^{3}}\right)^{5}$$
Simplify each expression, if possible.$$\left(\frac{u^{4}}{v^{2}}\right)^{6}$$
Simplify each expression, if possible.$$y^{3} y^{2} y^{4}$$
Simplify each expression, if possible.$$y^{4} y y^{6}$$
Simplify each expression, if possible.$$\frac{15^{9}}{15^{6}}$$
Simplify each expression, if possible.$$\frac{25^{13}}{25^{7}}$$
Simplify each expression, if possible.$$\frac{t^{5} t^{6} t}{t^{2} t^{3}}$$
Simplify each expression, if possible.$$\frac{m^{5} m^{12} m}{m^{7} m^{4}}$$
Simplify each expression, if possible.$$\frac{(k-2)^{15}}{(k-2)}$$
Simplify each expression, if possible.$$\frac{(m+8)^{20}}{(m+8)}$$
Simplify each expression, if possible.$$c d^{4} \cdot c d$$
Simplify each expression, if possible.$$a b^{3} \cdot a b^{4}$$
Simplify each expression, if possible.$$\left(\frac{y^{3} y^{5}}{y y^{2}}\right)^{3}$$
Simplify each expression, if possible.$$\left(\frac{s^{5} s^{6}}{s^{2} s^{2}}\right)^{4}$$
Simplify each expression, if possible.$$\frac{s^{2} s^{2} s^{2}}{s^{3} s}$$
Simplify each expression, if possible.$$\frac{w^{4} w^{4} w^{4}}{w^{2} w}$$
Simplify each expression, if possible.$$\left(-6 a^{3} b^{2}\right)^{3}$$
Simplify each expression, if possible.$$\left(-10 r^{3} s^{2}\right)^{2}$$
Simplify each expression, if possible.$$\left(\frac{3 m^{4}}{2 n^{5}}\right)^{5}$$
Simplify each expression, if possible.$$\left(\frac{2 s^{2}}{3 t^{5}}\right)^{5}$$
Simplify each expression, if possible.$$\frac{\left(a^{2} b^{2}\right)^{15}}{(a b)^{9}}$$
Simplify each expression, if possible.$$\frac{\left(s^{3} t^{3}\right)^{4}}{(s t)^{2}}$$
Simplify each expression, if possible.$$\left(n^{4} n\right)^{3}\left(n^{3}\right)^{6}$$
Simplify each expression, if possible.$$\left(y^{3} y\right)^{2}\left(y^{2}\right)^{2}$$
Simplify each expression, if possible.$$\frac{(6 h)^{8}}{(6 h)^{6}}$$
Simplify each expression, if possible.$$\frac{(-7 r)^{10}}{(-7 r)^{8}}$$
Simplify each expression, if possible.$$\frac{x^{4} y^{7}}{x y^{3}}$$
Simplify each expression, if possible.$$\frac{p^{7} q^{10}}{p^{2} q^{7}}$$
Simplify each expression, if possible.$$\left(\frac{m}{3}\right)^{4}$$
Simplify each expression, if possible.$$\left(\frac{n}{5}\right)^{3}$$
Look Alikes...A. $a^{3} \cdot a^{3}$B. $\left(a^{3}\right)^{3}$C. $a^{3}+a^{3}$
Look Alikes...A. $\left(m^{5}\right)^{7}$B. $m^{5} \cdot m^{7}$C. $m^{5}-m^{7}$
Look Alikes...A. $b^{3} b^{2} b^{4}$B. $\left(b^{3} b^{2}\right)^{4}$C. $\frac{b^{3} b^{2}}{b^{4}}$
Look Alikes...A. $\left(2 n^{4} n\right)^{5}$B. $2 n^{4} n$C. $\left(\frac{2 n^{4}}{n}\right)^{5}$
APPLICATIONSArt History. Leonardo da Vinci's drawing relating a human figure to a square and a circle is shown. Find an expression for the following:A. The area of the squareif the man's height is $5 x$ feetB. The area of the circleif the waist-to-feet distance is $3 a$ feet.Leave $\pi$ in youranswer.(PICTURE NOT COPY)
APPLICATIONSA bowling ball fits tightly against all sides of a cardboard box that it is packaged in. Find expressions for the volume of the ball and box. Leave $\pi$ in your answer.
Childbirth. Mr. and Mrs. Emory Harrison, of Johnson City, Tennessee, had 13 sons in a row during the 1940 s and 1950 s. The probability of a family of 13 children all being male is $\left(\frac{1}{2}\right)^{13} .$ Evaluate this expression.
APPLICATIONSA Super Ball is dropped from a height of 1 foot and always rebounds to four-fifths of its previous height. The rebound height of the ball after the third bounce is $\left(\frac{4}{5}\right)^{3}$ feet. Evaluate this expression. Is the third bounce more or less than $\frac{1}{2}$ foot high?
WRITINGExplain the mistake in the following work.A.(EQUATION NOT COPY)B.(EQUATION NOT COPY)
WRITINGExplain why we can simplify $x^{4} \cdot x^{5},$ but cannot simplify $x^{4}+x^{5}$
Match each equation with its graph below.$$y=2 x-1$$
Match each equation with its graph below.$$y=3 x-1$$
Match each equation with its graph below.$$y=3$$
Match each equation with its graph below.$$x=3$$
CHALLENGE PROBLEMSSimplify each expression. The variables represent natural numbers.A. $x^{2 m} x^{3 m}$B. $\left(y^{5}\right)^{4}$C. $\frac{m^{8 x}}{m^{4 x}}$D. $\left(2 a^{6 y}\right)^{4}$
CHALLENGE PROBLEMSEvaluate the following expression without using a calculator:$$\frac{(108,642)^{4}}{(54,321)^{4}}$$
