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Chapter 2: Problem 19

Simplify the given expression by writing it as a power of a single variable. $$ t^{4}\left(t^{3}\left(t^{-2}\right)^{5}\right)^{4} $$

Short Answer

Expert verified
The simplified expression is \(t^{-24}\).

Step by step solution

01

Apply the Power Rule

First, let's apply the power rule to (t^{-2})^5. The power rule states that (a^b)^c = a^(b*c), so we have: \( (t^{-2})^5 = t^{-2*5} = t^{-10} \) Now, substitute this back into the original expression: \( t^4 (t^3 (t^{-10})^1 )^4 \)
02

Simplify Inside the Parentheses

Next, let's simplify the expression inside the parentheses: \( t^3 * t^{-10} \) Using the product rule, which states that a^b * a^c = a^(b+c), we get: \( t^{3+(-10)} = t^{-7} \) Now, substitute this back into the expression: \( t^4 (t^{-7})^4 \)
03

Apply the Power Rule again

Now, apply the power rule to (t^{-7})^4. Using the power rule, (a^b)^c = a^(b*c), we have: \( (t^{-7})^4 = t^{-7*4} = t^{-28} \) Now, substitute this back into the expression: \( t^4 * t^{-28} \)
04

Final Simplification

Finally, simplify the expression using the product rule: \( t^4 * t^{-28} = t^{4 + (-28)} = t^{-24} \) The simplified expression is written as a power of a single variable: \( t^{-24} \).

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Most popular questions from this chapter

Write the indicated expression as \(a\) polynomial. $$ (q(x))^{2} s(x) $$

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Write the indicated expression as \(a\) polynomial. $$ (p(x))^{2} s(x) $$
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