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Chapter 8: Problem 29

Simplify each expression. SEE EXAMPLE \(2 .\) (OBJECTIVE 2 ) $$1,000^{2 / 3}$$

Short Answer

Expert verified
The simplified form of the expression \(1,000 ^{2 / 3}\) is 100

Step by step solution

01

Convert the Fractional Exponent into a Root

We convert the fraction exponent \(2 / 3\) to cubic root since the denominator of the fractional exponent represents the root. So, \(1,000 ^{2 / 3}\) can be rewritten as \( \sqrt [3]{1000^2}\)
02

Square the number under the cubic root

Now, find the square (second power) of 1000 which is \(1000 * 1000 = 1,000,000\) . So, our expression becomes \( \sqrt [3]{1,000,000}\)
03

Find the Cubic Root

The next step is to find the cubic root of 1,000,000. The cubic root of a number a is the number x such that \(x^3 = a\). In this case, \(x^3 = 1,000,000\), and by calculating, we find out that \(x = 100\)

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Key Concepts

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

Fractional Exponents
In mathematics, fractional exponents are a way to express roots as exponents. They are a powerful tool for simplifying calculations involving roots and powers. When you see a number raised to a fractional exponent, for example, \(a^{m/n}\), it indicates the \(n\)-th root of \(a\) raised to the \(m\)-th power.
Here's how it works:
  • The numerator of the fraction \(m/n\) is the power you raise \(a\) to.
  • The denominator \(n\) denotes the root, which means you take the \(n\)-th root of \(a\).
For example, \(1,000^{2/3}\) represents the cube root of 1,000 squared. Understanding this transformation is crucial for simplifying expressions involving fractional exponents.
Cubic Roots
Cubic roots are a specific type of root and are used when you need to find a number that, when cubed, gives the original number. In general, the cube root of a number \(a\) is written as \( \sqrt[3]{a} \) and solves the equation \(x^3 = a\).
To calculate the cube root:
  • Determine a number \(x\) such that when multiplying \(x\) three times (\(x \times x \times x\)), you get \(a\).
  • For example, if you calculate \(\sqrt[3]{1,000,000}\), you're looking for the number which when cubed equals 1,000,000.
The cube root is useful when working with volume-related problems or simplifying expressions with cubic powers involved.
Simplifying Expressions
Simplifying expressions involves reducing mathematical expressions into their simplest form, which makes them easier to handle and solve. When you simplify, you perform operations that help identify equivalent expressions while ensuring they retain their meaning.
Here's how to approach simplification with roots and powers:
  • First, convert any roots into fractional exponents if necessary to utilize exponent rules more effectively.
  • Apply exponent rules such as the power of a power rule, where \((a^m)^n = a^{m \times n}\).
    This helps break down the complexity of combined operations.
  • Perform arithmetic calculations like multiplication or finding roots step by step.
During simplification, keeping track of the sequence of operations is vital. Ensuring each step follows logically from the previous one is crucial when aiming for the final simplified form.

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Most popular questions from this chapter

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\left(3^{2 / 3} 5^{1 / 3}\right)^{3}$$

Explain how to rationalize a denominator.

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(\frac{9}{64}\right)^{1 / 2}$$

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(\frac{1}{4}\right)^{1 / 2}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$(-27)^{-4 / 3}$$
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