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Chapter 1: Problem 7

Express the number in the form a/b, where \(a\) and \(b\) are integers. $$9^{5 / 2}$$

Short Answer

Expert verified
\(9^{5/2} = \frac{243}{1}\).

Step by step solution

01

Understand the Expression

The expression given is \(9^{5/2}\). The fraction \(\frac{5}{2}\) in the exponent can be interpreted as \(9^{5/2} = (9^{1/2})^5\). This means we find the square root of 9 and then raise the result to the 5th power.
02

Calculate the Square Root

The square root of 9 is calculated as \(9^{1/2} = \sqrt{9} = 3\). So, \(9^{1/2} = 3\).
03

Raise to the 5th Power

Now, we take the result from Step 2 and raise it to the 5th power: \((3)^5 = 3 \times 3 \times 3 \times 3 \times 3\).
04

Compute Exponentiation

Calculating \(3^5\), we multiply: \(3 \times 3 = 9\); then, \(9 \times 3 = 27\); next, \(27 \times 3 = 81\); finally, \(81 \times 3 = 243\). Thus, \(3^5 = 243\).
05

Write as a Fraction

Since the result \(243\) is an integer, it can be expressed as a fraction where the denominator is 1. Therefore, the expression \(9^{5/2}\) can be written as \(\frac{243}{1}\).

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Key Concepts

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

Rational Exponents
Rational exponents involve the use of fractions in the exponent of a base number. The expression \(9^{5/2}\) is a good example of this. Here, the fraction \(\frac{5}{2}\) acts as the exponent.
This exponent can be seen as combining two operations: taking the root and raising to a power.
  • The denominator (2) represents the type of root, in this case, a square root, because \( \frac{1}{2} \) corresponds to a square root.
  • The numerator (5) indicates the power to which the result should be raised.
By understanding rational exponents, we can rewrite the expression \(9^{5/2}\) as \((9^{1/2})^5\). This shows how to take the problem step by step and simplifies the complexity of handling such expressions.
Square Root
The square root is an essential concept when dealing with rational exponents. A square root asks the question: what number, when multiplied by itself, will give the original number? For instance, the square root of 9, written as \( \sqrt{9} \), asks which number multiplied by itself equals 9.
The answer is 3, because \(3 \times 3 = 9\).
  • For practical use, the square root symbol \( \sqrt{} \) is often replaced by the exponent form \(9^{1/2}\).
  • This transformation from root notation to exponent notation is crucial for carrying out calculations further, especially in exponential form.
  • Knowing this allows us to rewrite expressions like \( (\sqrt{9})^5 \) seamlessly into the format \((3)^5\).
Fraction Representation
Once the exponentiation process is complete, it is important to represent the result clearly. In mathematical terms, expressing a number in the form \( \frac{a}{b} \) where both \( a \) and \( b \) are integers provides clarity.
For the problem \(9^{5/2}\), as calculated in the original solution, this simplifies to an integer, 243.
  • When a result is an integer like 243, it can simply be expressed as \( \frac{243}{1} \).
  • Formulating numbers in this fraction form helps emphasize the integer nature, especially when dealing with expressions containing exponents.
  • It also makes calculations consistent, following a standardized format.
Understanding fraction representation ensures that even exponential results remain manageable and understandable.

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Most popular questions from this chapter

Simplify the expression. $$\frac{2 x}{x+2}-\frac{8}{x^{2}+2 x}+\frac{3}{x}$$

Simplify the expression. $$\frac{\frac{b}{a}-\frac{a}{b}}{\frac{1}{a}-\frac{1}{b}}$$

Simplify the expression. $$\frac{(x+h)^{3}+5(x+h)-\left(x^{3}+5 x\right)}{h}$$

Rationalize the denominator. $$\frac{16 x^{2}-y^{2}}{2 \sqrt{x}-\sqrt{y}}$$

Simplify the expression. $$\frac{\frac{x+2}{x}-\frac{a+2}{a}}{x-a}$$
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