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Chapter 3: Problem 7

Evaluate (a) \(4^{1 / 2}\) (b) \(16^{3 / 4}\) (c) \(27^{2 / 3}\) (d) \(9^{-1 / 2}\)

Short Answer

Expert verified
(a) 2; (b) 8; (c) 9; (d) 1/3.

Step by step solution

01

Understanding Radicals in Exponent Form

To evaluate expressions like these, remember that fractional exponents represent radicals. The expression \(a^{m/n}\) can be interpreted as \(\sqrt[n]{a^m}\). This means the numerator \(m\) is a power and the denominator \(n\) is a root.
02

Evaluating (a) \(4^{1/2}\)

Start by recognizing that \(4^{1/2}\) represents the square root of 4. Thus, \(4^{1/2} = \sqrt{4}\). Calculate the square root: \(\sqrt{4} = 2\).
03

Evaluating (b) \(16^{3/4}\)

First, recognize \(16^{3/4}\) means the fourth root of 16, raised to the power of 3. Calculate the fourth root of 16 first: \(\sqrt[4]{16} = 2\), since \(2^4 = 16\). Then raise 2 to the power of 3: \(2^3 = 8\). Therefore, \(16^{3/4} = 8\).
04

Evaluating (c) \(27^{2/3}\)

This expression means find the cube root of 27, then square the result. \(\sqrt[3]{27} = 3\) because \(3^3 = 27\). Then square this result: \(3^2 = 9\). Hence, \(27^{2/3} = 9\).
05

Evaluating (d) \(9^{-1/2}\)

Here, \(9^{-1/2}\) indicates the reciprocal of the square root of 9. Calculate the square root: \(\sqrt{9} = 3\). Therefore, the reciprocal is \(1/3\). Hence \(9^{-1/2} = 1/3\).

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Key Concepts

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

Radicals
Radicals are a way to represent roots in mathematics. When you see a symbol like \( \sqrt[n]{a} \), it indicates the nth root of a number \( a \). This means you are looking for a number which, when multiplied by itself \( n \) times, gives you \( a \). Radicals can be used to express higher roots like cube roots, fourth roots, and so on.
  • When \( n = 2 \), it is called the square root.
  • When \( n = 3 \), it is called the cube root.
Understanding radicals makes it easier to evaluate expressions with fractional exponents. In these expressions, the base number is taken to a fractional power, where the denominator of the fraction indicates the root taken, while the numerator indicates the power it is raised to.
Square Root
The square root is a specific radical representing the number that, when multiplied by itself, results in the original number. For the expression \( a^{1/2} \), it refers to this operation.
  • Example: \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).
  • Fractional exponent form: \( 4^{1/2} = \sqrt{4} \).
Square roots are foundational in mathematical operations, often used to simplify expressions or solve equations. They have numerous applications in geometry, trigonometry, and calculus.
Cube Root
The cube root involves finding a number which, when used three times in a multiplication, returns the original number. Equivalent fractional exponent representation is \( a^{1/3} \). This operation is crucial for simplifying expressions involving third powers.
  • Example: \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \).
  • Fractional exponent form: \( 27^{1/3} = \sqrt[3]{27} \).
Cube roots can be found geometrically as well, representing the side length of a cube with a certain volume. Mastering cube roots helps in complex algebraic manipulations and in understanding three-dimensional geometry.
Reciprocal
The concept of a reciprocal is fundamental in understanding negative exponents. The reciprocal of a number is simply \( 1 \) divided by that number. For instance, the reciprocal of \( 3 \) is \( 1/3 \).
  • When dealing with negative exponents, like \( 9^{-1/2} \), it implies the operation involves a reciprocal.
  • Example: \( 9^{-1/2} = \frac{1}{\sqrt{9}} = 1/3 \).
Reciprocals underline the inverse relationship in multiplication and division. They are quite useful in conversions and comparisons, key for solving equations and understanding the balance of mathematical expressions.

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Most popular questions from this chapter

Simplify $$ \frac{\left(\frac{4}{3}\right)^{3} \times\left(\frac{3}{5}\right)^{-2}}{\left(\frac{2}{5}\right)^{-3}} $$ giving the answer with positive indices.

Evaluate (a) \(\left(3.75 \times 10^{3}\right)\left(6 \times 10^{4}\right)\) and (b) \(\frac{3.5 \times 10^{5}}{7 \times 10^{2}}\) expressing answers in standard form.

Find the value of: (a) \(\frac{7^{5}}{7^{3}}\) and (b) \(\frac{5^{7}}{5^{4}}\)

Express the following numbers, given in standard form, as fractions or mixed numbers: (a) \(2.5 \times 10^{-1}\) (b) \(6.25 \times 10^{-2}\) (c) \(1.354 \times 10^{2}\)

Simplify \(\frac{7^{-3} \times 3^{4}}{3^{-2} \times 7^{5} \times 5^{-2}}\), expressing the answer in index form with positive indices.
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