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Chapter 8: Problem 6

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$64^{\frac{1}{3}}$$

Short Answer

Expert verified
The simplified form of \(64^{\frac{1}{3}}\) or \(\sqrt[3]{64}\) is 4.

Step by step solution

01

Convert Exponential Form to Radical Form

The given expression is in exponent form \(64^{\frac{1}{3}}\). To convert this into its radical form, the divisor of the fraction, which is 3 in this case, becomes the index of the radical. Thus, the cubic root of 64 is written as \(\sqrt[3]{64}\).
02

Simplify the Radical

The next step is to simplify the cubic root of 64. A cubic root of a number is the value that when cubed gives the original number. In simpler terms, what number by itself three times equals 64? The cubic root of 64 is 4 because \(4^3=64\).
03

Verification with a Calculator

The final step is to check the work using a calculator. Verify that \(64^{\frac{1}{3}}\) or \(\sqrt[3]{64}\) indeed equals 4. A calculator should confirm this.

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Key Concepts

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

Exponentiation
Exponentiation is a method involving numbers where a base is raised to a power. In the expression \(64^{\frac{1}{3}}\), 64 is the base, and \(\frac{1}{3}\) is the exponent. The exponent tells us how many times the base is multiplied by itself. When the exponent is a fraction, it indicates a root operation. For instance, an exponent like \(\frac{1}{3}\) signifies a cubic root. This is because the exponent's numerator is the power and the denominator is the root.
  • An exponent of \(1\) means the number remains unchanged.
  • An exponent that is a whole number is straightforward: multiply the base by itself as many times as the exponent indicates.
  • A fractional exponent suggests a root operation instead of mere multiplication.
Understanding exponentiation is essential for simplifying expressions efficiently.
Cubic Root
A cubic root finds the number that, when multiplied by itself three times, results in the given value. For \(64^{\frac{1}{3}}\), converting this expression to a radical form involves substituting the exponent \(\frac{1}{3}\) with a cube root symbol, which results in \(\sqrt[3]{64}\).
  • Cubic roots are opposite to cubing, which is sharing a number by double-multiplying.
  • The cubic root is unique because not many numbers when cubed will produce another integer.
In this case, the cubic root of 64 is 4 because if you multiply 4 by itself three times, you get 64 \((4 \times 4 \times 4 = 64)\). This is a particularly useful operation in algebra and physics where these roots occur frequently.
Simplification
Simplification is the process of reducing a mathematical expression to its simplest form. The goal is to make the expression easier to understand and work with. For example, simplifying \(\sqrt[3]{64}\) results in finding the cube root which is 4.
Here are some steps to simplify radical expressions:
  • Identify and rewrite in radical form if the expression is in exponential form.
  • Calculate the root, using known values whenever possible, like \(\sqrt[3]{64} = 4\).
  • Ensure complete reduction, often requiring factoring in more complex expressions.
Simplification helps to streamline complex calculations, making them easier to follow and solve.
Calculator Verification
Calculator verification is a crucial step to ensure that your manual calculations are correct. Once you've simplified your expression by hand, it's good practice to use a calculator for confirmation.
To verify \(64^{\frac{1}{3}}\) using a calculator:
  • Switch the calculator to handle exponents correctly, ensuring you can input fractional exponents.
  • Calculate \(64^{\frac{1}{3}}\), which should yield a result of 4.
  • Cross-check that you inputted everything correctly during manual processing.
By doing this, you gain confidence that you've reached the right solution, catching errors in prior steps that could have slipped past. Using this method strengthens computational accuracy in more complex mathematical tasks.

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Most popular questions from this chapter

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$625^{-\frac{3}{4}}$$

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{\sqrt{2}}{\sqrt{3}}+\frac{\sqrt{3}}{\sqrt{2}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$25^{-\frac{1}{2}}=-5$$

Simplify each expression. Write answers in exponential form with positive exponents only. Assume that all variables represent positive real numbers. $$\left(\frac{x^{\frac{2}{5}}}{x^{\frac{6}{5}} \cdot x^{\frac{3}{5}}}\right)^{5}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}=\left(\frac{1}{4}\right)^{-1}$$
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