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Chapter 1: Problem 43

Simplify. \(9^{3 / 2}\)

Short Answer

Expert verified
Simplified result is 27.

Step by step solution

01

Understand the Exponent

The exponent \(\frac{3}{2}\) means that we need to apply both root and power operations to the base number 9.
02

Apply the Root Operation

The fraction in the exponent can be interpreted as \(9^{(3)(1/2)}\). The expression \(1/2\) indicates a square root. So, we first find the square root of 9, which is \(\sqrt{9} = 3.\)
03

Apply the Power Operation

Now, raise the result from the previous step to the power of 3. So, we calculate \(3^3\): \(3^3 = 27.\)

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Key Concepts

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

Rational Exponents
Rational exponents may look complicated, but they offer a useful way to write roots and powers using fractions. An exponent like \ \( \frac{a}{b} \ \) means you first take the root of the number and then raise it to a power. In the exercise, \ \( 9^{3 / 2} \ \), the exponent \ \( \frac{3}{2} \ \) suggests applying these steps.
Let鈥檚 break this down: \ \( 9^{\frac{3}{2}} = (\sqrt{9})^3 = 3^3 = 27 \ \). Here are the steps:
  • Interpret the denominator as the root. The exponent's denominator (2) signifies a square root.
  • Apply the square root operation first, which results in 3.
  • Next, use the numerator (3) as the exponent to raise the result to the power. Hence, \ \( 3^3 = 27 \ \).
This method can simplify concepts involving roots and powers greatly, making calculations easier to manage.
Square Root
When you see an expression like \ \( \sqrt{x} \ \), it refers to the square root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example:
  • \ \( \sqrt{9} = 3 \ \), because \ \( 3 \times 3 = 9 \ \).
Using square roots can greatly simplify expressions with exponents. In the step-by-step solution, to simplify \ \( 9^{3/2} \ \), the square root operation is a key aspect:
  • First, the fraction in the exponent was interpreted to find \ \( 9^{(3)(1/2)} \ \).
  • The square root of 9 was calculated as \ \( 3 \ \).
By understanding and identifying square roots, you can make sense of more complex fractional exponents and simplify calculations more effectively.
Power Operations
Power operations involve raising a number to a specific exponent. This means multiplying the number by itself a certain number of times. For example:
  • \ \( 2^3 = 2 \times 2 \times 2 = 8 \ \).
  • \ \( 4^2 = 4 \times 4 = 16 \ \).
In the exercise, we simplified \ \( 9^{3/2} \ \) by performing power operations in two steps:
  • First, determine the square root as \ \( 3 \ \).
  • Then, raise the result (3) to the power specified by the numerator (3). Hence, \ \( 3^3 = 27 \ \).
Understanding power operations allows you to handle both the components of the fraction in the exponent: the root (denominator) and the power (numerator). This clarity simplifies the evaluation process and makes exponentiation more intuitive.

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