Rewrite the expression using exponent rules. ( 4 x 2 ) 3 \left(4x^2\right)^3 ( 4 x 2 ) 3

Hey, everyone. So in this problem, we're gonna rewrite the expression using the exponent rules that we've discussed so far. Let's take a look. So we have 4 x to the second power, and that whole entire expression is raised to the third power. So what are the rules which one of the rules am I gonna use here? I see an exponent that's on the outside of a parenthesis, and usually that means that we're gonna use some of the power rules. So it turns out what happens here is we have a product. We have 4, that's one term, and then the x squared, that's another term. So we have to do is we have to distribute this exponent into everything that's inside of the parentheses. So this whole thing here is the, if you remember, it's the power of a product. So it's the power of a product rule. Hopefully, you have your rule sheet handy. You're just gonna distribute the 3rd 3rd exponent into the 4, so that becomes 4 cubed, and then you're gonna do the x squared, and then that's gonna be cubed as well. So are we done? Well, no, because now what happens is I have a power that's outside of a power. So I have a power over product, that's what I did first. And then for this situation over here, this is actually just the normal power rule. So the power rule says that when you have powers that are outside of powers, then you multiply their exponents. So this just becomes 4 cubed, and then this becomes x to the 2 times 3 power. So now what we can do is we can just evaluate. 4 cubed is 4 times 4, which is 16, and then 16 times 4 is 64, so that's what that becomes. And then what is x to the 2 times 3 power? That's just x to the 6th power. So that's the simplest you can write this expression, and we had to expand using the power rules. Hopefully, that made sense. Thanks for watching.

0. Functions

Exponent Rules

Business Calculus

0. Functions

Exponent Rules

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Multiple Choice

Rewrite the expression using exponent rules.
$\left(4x^2\right)^3$

$64x^6$

$64x^5$

$4x^6$

$64x^3$

Verified step by step guidance

Step 1: Recognize that the given expression is $(4x^2)^3$. This involves applying the power rule of exponents, which states $(a^m)^n = a^{m \cdot n}$.

Step 2: Break the expression into its components. Here, $4$ and $x^2$ are being raised to the power of 3. This means we can rewrite the expression as $4^3 \cdot (x^2)^3$.

Step 3: Simplify $4^3$. Using the rule for exponents, $4^3 = 4 \cdot 4 \cdot 4$.

Step 4: Simplify $(x^2)^3$ using the power rule of exponents. Multiply the exponents: $2 \cdot 3 = 6$, so $(x^2)^3 = x^6$.

Step 5: Combine the results. The simplified expression is $4^3 \cdot x^6$, which evaluates to $64x^6$.

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Struggling with Business Calculus?

Rewrite the expression using exponent rules.(4x2)3\left(4x^2\right)^3(4x2)3

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Exponent Rules practice set

Rewrite the expression using exponent rules.
$\left(4x^2\right)^3$