Simplify. Assume all variables represent nonzero real numbers. The answer should not contain neg

step 1 as to simplify the given expression. And there are multiple ways to do this. However, the first

Simplify. Assume all variables represent nonzero real...

Key Concepts

Negative Exponent Rule

The negative exponent rule states that for any nonzero number a and positive integer n, a^(-n) is equal to 1/(a^n). This rule is fundamental when rewriting expressions so that all exponents are positive, which is a common requirement in algebraic simplification.

Power of a Product Rule

The power of a product rule explains that when a product of terms is raised to an exponent, each factor in the product may be raised to that exponent individually. This means that (abc)^n equals a^n * b^n * c^n, which simplifies the process of expanding and reducing expressions involving products.

Power of a Power Rule

The power of a power rule dictates that when an exponentiated expression is raised to another exponent, the exponents multiply. In other words, (a^m)^n is equal to a^(m*n). This rule is essential for simplifying expressions where powers are nested.

Product of Powers Rule

The product of powers rule states that when multiplying two exponents with the same base, their exponents can be added, so a^m * a^n equals a^(m+n). This rule helps combine terms and simplify expressions that have the same base raised to different powers.

Quotient of Powers Rule

The quotient of powers rule indicates that when dividing two expressions with the same base, one can subtract the exponent of the denominator from the exponent of the numerator. Specifically, a^m / a^n equals a^(m-n). This rule is key in simplifying rational expressions by combining like bases.

Transcript

00:01 As to simplify the given expression.

00:02 And there are multiple ways to do this.

00:05 However, the first thing we need to do is apply those exponents on the outside.

00:09 So the first thing in the numerator, i have to raise each of our terms to the negative fifth power.

00:15 So in other words, i'll have two to the negative fifth power.

00:19 We'll raise you to the negative fifth to the negative fifth.

00:23 We'll take v squared, raise it to the negative fifth, and w4 also needs to be raised to the negative fifth.

00:29 The same idea has to the negative fifth.

00:31 The same idea, happens to the denominator.

00:32 We have to raise each of these terms to the second power.

00:35 So in other words, we'll have u to the 6 power squared.

00:39 Then we're going to have to square v to the negative fifth, and then we're going to have to square w to the negative 10th.

00:46 All right, so let's go ahead and do this.

00:47 Well, first, we have 2 to the negative fifth.

00:50 So to get rid of that negative exponent, i need to take the reciprocal.

00:54 In other words, that would give me 1 over 2 to the 5th power.

00:59 Now, for the rest of these, we can apply our power rule, which says when we take a base to a power and raise it to a number one, all we do is multiply the exponents.

01:09 So for u, we're going to multiply negative 5 times negative 5, which is positive 25.

01:15 For v, we're going to multiply 2 times negative 5, which is negative 10.

01:20 For w, 4 times negative 5 is negative 20.

01:24 The same thing's going to happen in the denominator...

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