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Chapter 0: Problem 36

Use the laws of exponents to compute the numbers. \(20^{-5} \cdot 5^{5}\)

Short Answer

Expert verified
\( \frac{1}{2^{10}} \)

Step by step solution

01

Express both numbers with the same base

First, notice that 20 and 5 can be written as powers of 5. We know that 20 = 4 \times 5, which can be written as 20 = 2^2 \times 5. Therefore, we can rewrite 20^{-5} as \( (2^2 \times 5)^{-5} \).
02

Apply the power of a product rule

Use the power of a product rule, which states that \( (ab)^n = a^n \times b^n \), to the expression \( (2^2 \times 5)^{-5} \). This gives us \( 2^{2 \times (-5)} \times 5^{-5} = 2^{-10} \times 5^{-5} \).
03

Combine the exponents

We now have \( 2^{-10} \times 5^{-5} \times 5^5 \). Using the product of powers property, which is \( a^m \times a^n = a^{m+n} \), we combine \( 5^{-5} \times 5^5 \) to get \( 5^{0} \). We know that any number to the power of zero is 1, so we have \( 2^{-10} \times 1 \).
04

Simplify the final expression

Simplify \( 2^{-10} \times 1 \). Since multiplying by 1 does not change the value, the final simplified expression is just \( 2^{-10} \).
05

Rewritten expression

The number \( 2^{-10} \) can be written as \( \frac{1}{2^{10}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Product Rule
When dealing with exponents, understanding the power of a product rule can simplify many expressions. This rule tells us that \( (ab)^n = a^n \backslash\times b^n \). This means if we have a product of two bases raised to a power, we can distribute the power to each base individually. For instance, looking at the exercise, we transformed \( (2^2 \backslash\times 5)^{-5} \) into \( 2^{2 \backslash\cdot (-5)} \backslash\times 5^{-5} \).
Breaking it down:
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Most popular questions from this chapter

Let \(f(x)=x^{2}+3 x+1\) and let \(g(x)=x^{2}-3 x-1\). Graph the two functions \(f(g(x))\) and \(g(f(x))\) together in the window \([-4,4]\) by \([-10,10]\) and determine if they are the same function.

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