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Chapter 10: Problem 38

Which is greater? \(\left(4^{4}\right)^{5}\) or \(\left(4^{5}\right)^{4}\)

Short Answer

Expert verified
Both are equal: \( \big(4^4\big)^5 = \big(4^5\big)^4 = 4^{20}\).

Step by step solution

01

Understand the Expression

Identify the two expressions that need to be compared: \ \(a = \big(4^4\big)^5\) and \(b = \big(4^5\big)^4 \).
02

Apply the Power of a Power Rule

Use the exponentiation rule \(\big(a^m\big)^n = a^{mn}\), which allows combining exponents. Apply this to both expressions: \(a = 4^{4*5} = 4^{20}\) and \(b = 4^{5*4} = 4^{20}\).
03

Compare the Results

Both expressions result in the same value: \(a = 4^{20}\) and \(b = 4^{20}\). Hence, \(a = b\).

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Key Concepts

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

Exponentiation Rules
Exponentiation involves raising a number, known as the base, to a power, which is the exponent. There are several key rules for working with exponents:
  • Product of Powers Rule: When multiplying two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Power of a Power Rule: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n}\).
  • Power of a Product Rule: When raising a product to a power, you apply the exponent to each factor: \( (ab)^n = a^n \times b^n\).
Understanding these rules is essential for simplifying and comparing expressions involving exponents, as seen in the given exercise.
Exponential Comparison
Comparing exponential expressions involves using exponentiation rules to simplify the expressions into a common form. This was illustrated in the exercise comparing \( \big(4^4\big)^5\) and \( \big(4^5\big)^4 \). First, we applied the Power of a Power Rule:
\[ \big(4^4\big)^5 = 4^{4 \times 5} = 4^{20} \]
\[ \big(4^5\big)^4 = 4^{5 \times 4} = 4^{20} \]
As you can see, both expressions simplify to \( 4^{20} \). Thus, comparing them becomes straightforward since they are equal.
When comparing different exponential forms, it is crucial to reduce them to a common base and exponents to see similarities or differences easily.
Power of a Power Rule
The Power of a Power Rule states that when you have an exponent raised to another exponent, you multiply the exponents. For example, \( (a^m)^n = a^{m \times n} \). This rule simplifies complex exponential expressions.

In the given exercise:
\[ \big(4^4\big)^5 = 4^{4 \times 5} = 4^{20} \]
\[ \big(4^5\big)^4 = 4^{5 \times 4} = 4^{20} \]
Both expressions simplify to the same form, making them easy to compare.
Understanding this rule helps recognize patterns and relationships in exponential expressions, simplifying problem-solving processes in algebra and prealgebra.

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Most popular questions from this chapter

Simplify. $$\left(p^{44} z^{32}\right)^{3}$$

Multiply the polynomials. $$\begin{array}{r}y^{2}-7 y+3 \\\\\times \quad 3 y-5 \\\\\hline\end{array}$$

Perform the indicated operations. $$\left(2 x^{3}-5 x+8\right)-\left(4 x^{2}+2 x-3\right)+\left(-7 x^{3}+8 x^{2}\right)$$

$$\text { Fill in the blanks }\left(x^{3}\right)^{5}=x \square \cdot \square=x^{15}$$

Perform the indicated operations. $$\left(-9 m^{4}-7 m^{2}+8\right)-\left(3 m^{4}+9 m^{3}-11\right)-\left(8 m^{3}+m^{2}-3\right)$$
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