Write each expression using a single exponent. a. (5^3)/(5^6) b. (14^3)^6 c. 8^3 ·8^6 d. (16^6)/

step 1 We need to write each expression using a single exponent. So the first one, since we have the sa

Write each expression using a single exponent. a....

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Key Concepts

Laws of Exponents

The laws of exponents are a set of rules that allow the simplification and manipulation of expressions involving powers. They help when multiplying, dividing, or raising a power to another power, ensuring that calculations can be streamlined into a single exponent form.

Product of Powers Rule

When multiplying expressions with the same base, you add the exponents. Mathematically, for any nonzero number a, a^m * a^n equals a^(m+n), allowing the expression to be rewritten as a single power.

Quotient of Powers Rule

When dividing expressions with the same base, the exponents are subtracted. For any nonzero a, a^m / a^n is simplified to a^(m-n), which consolidates the expression into one exponent.

Power of a Power Rule

When an exponentiated expression is raised to another power, the exponents multiply. Specifically, (a^m)^n equals a^(m*n), turning complex exponent expressions into a simpler single exponent form.

Negative Exponent Rule

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. That is, a^(-n) is equivalent to 1/(a^n), which helps in rewriting expressions so that all exponents are positive.

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