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Chapter 7: Problem 26

Evaluate each exponential. $$ 64^{3 / 2} $$

Short Answer

Expert verified
512

Step by step solution

01

Understand the Exponent Fraction

The exponent \(\frac{3}{2}\) indicates both an operation of square rooting and cubing the number. That is, the number is taken to the power of 3, and then the square root is taken, or vice versa.
02

Rewrite using Radicals

Rewrite the expression \(64^{3/2}\) using a radical form: \(\big(64^{1/2}\big)^3\). This means we will first compute the square root of 64, then raise the result to the third power.
03

Compute the Square Root

Find the square root of 64. Recall that the square root of 64 is 8, because \(8^2 = 64\). So, \(64^{1/2} = 8\).
04

Raise to the Power of 3

Now, take the result from the previous step and raise it to the power of 3: \(8^3\). Compute \(8^3 = 8 \times 8 \times 8 = 512\).
05

Combine the Results

Combining all steps, we get the value of \(64^{3/2} = 512\).

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Key Concepts

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

Fractional Exponents
Fractional exponents, such as \(\frac{3}{2}\), represent both a root and a power. The numerator indicates the power, and the denominator indicates the root. In other words, \((a^{m/n})\) is equivalent to taking the n-th root of a raised to the m-th power. For example, \(64^{3/2}\) means we first find the square root of 64, and then raise the result to the 3rd power, or vice versa. This can be summarized as: \[64^{3/2} = (64^{1/2})^3 = 8^3 = 512.\]
Understanding the behavior of fractional exponents can simplify complex problems involving exponents and roots.
  • Numerator: Indicates the power
  • Denominator: Indicates the root
Learning these basics makes it easier to handle and evaluate expressions in various mathematical contexts.
Square Root
The square root is a common mathematical operation that returns a number which, when multiplied by itself, results in the original number. For example, the square root of 64 is 8 because \(8 \times 8 = 64\). The square root is denoted by the radical symbol \(\big(\big(\big(\big(\big(\big(\big(\big(\big(\sqrt{} \big(\big(\big(\big(\big(\big(\big(\big(\big(\sqrt{} \big(\big(\big(\big((\sqrt{} \big(\big(\big(\big((\sqrt{} \).\

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Most popular questions from this chapter

Rationalize each numerator. Assume that all variables represent positive real numbers. $$ \frac{2 \sqrt{x}-\sqrt{y}}{3 x} $$

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{3-3 \sqrt{5}}{3} $$

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{-5+5 \sqrt{2}}{5} $$

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$\sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b$$ Explain why this is faulty reasoning.

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{11 y-\sqrt{242 y^{5}}}{22 y} $$
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