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Chapter 1: Problem 3

Simplify (without your GDC) each expression to a single integer. $$64^{\frac{2}{3}}$$

Short Answer

Expert verified
The simplified expression is 16.

Step by step solution

01

Understand the expression

The expression given is \(64^{\frac{2}{3}}\). This represents a number being raised to a fractional power, which involves both a root and an exponentiation.
02

Apply the exponentiation rule

The expression 64 raised to the power \(\frac{2}{3}\) can be rewritten using the property \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m \). So, \(64^{\frac{2}{3}} = (\sqrt[3]{64})^2\).
03

Compute the cube root

Find the cube root of 64, which is \(\sqrt[3]{64}\). Since \(4^3 = 64\), the cube root of 64 is 4.
04

Raise the result to the second power

Now that we know \(\sqrt[3]{64} = 4\), we compute \(4^2\). Calculating this gives us \(16\).

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Key Concepts

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

Understanding Cube Root
Cube roots are essentially the opposite of cubing a number. They help us find a number that, when multiplied by itself three times, gives the original number. In the expression \(64^{\frac{2}{3}}\), the fraction \(\frac{2}{3}\) indicates the operation of finding a cube root first before dealing with the exponent. Here, \(\sqrt[3]{64}\) is asking "what number multiplied by itself three times results in 64?"
  • Calculating \(4 \times 4 \times 4 = 64\), we find that the cube root of 64 is 4.
  • This process is crucial when dealing with fractional exponents, as it breaks down into more manageable steps.
Decoding Fractional Exponents
Fractional exponents can sometimes look intimidating, but they actually simplify mathematical processes. They offer a way to express roots and powers in a single expression. When you see an expression like \(64^{\frac{2}{3}}\), it represents taking the cube root of 64 and then squaring the result.
  • The denominator (3) tells us the type of root to calculate, here a cube root.
  • The numerator (2) indicates the power to raise the result of the cube root.
By translating these fractional exponents into roots and powers, complex expressions can become much clearer.
Mastering Mathematical Simplification
Mathematical simplification involves reducing expressions to their most straightforward form. This is especially important in solving problems efficiently. In our example, simplifying \(64^{\frac{2}{3}}\) requires understanding the structure of fractional exponents.
  • First, we break it down: Calculate the cube root of 64, which is 4.
  • Next, raise this result to the power of 2: \(4^2 = 16\).
The goal is to transform the expression into a simpler integer. By dissecting the operation step by step, complex problems become easy to handle and understand.

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Most popular questions from this chapter

Evaluate each absolute value expression. $$\frac{-1}{|-1|}$$

Write an inequality to represent the given interval and state whether the interval is closed, open or half-open. Also state whether the interval is bounded or unbounded. $$]-10,-2]$$

Find all values of \(x\) that make the equation true. $$|x+5|=-2$$

Use the substitution method to solve each pair of simultaneous equations. $$\begin{aligned} &\frac{x}{5}+\frac{y}{2}=8\\\ &x+y=20 \end{aligned}$$

Solve the pair of simultaneous equations using any method 鈥揺limination, substitution or the graphical features of your GDC. $$\begin{aligned} &3 x+2 y=9\\\ &7 x+11 y=2 \end{aligned}$$
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