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Chapter 5: Problem 12

\(9^{3 / 2}=27\)

Short Answer

Expert verified
The given equation \(9^{3 / 2}=27\) is indeed correct.

Step by step solution

01

Understand the Fractional Exponent

A fractional exponent means that we have to take the square root of the base then raise the result to the power. In this particular case, we will start by taking the square root of 9.The square root of 9 is 3. The equation now becomes \(3^{3}=27\).
02

Evaluate \(3^{3}\)

We know that the cube of 3 is 27, so our equation becomes \(27=27\).
03

Verify the Result

From Step 2, the left-hand side (LHS) is equal to the right-hand side (RHS).

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Key Concepts

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

Square Roots
Square roots are mathematical operations that find a number which, when multiplied by itself, gives the original number. This is particularly useful when dealing with fractional exponents.
For example, the square root of 9 is 3 since \(3 imes 3 = 9\).
  • The symbol for square root is \( \sqrt{} \).
  • It is important in solving equations with fractional exponents, as seen in the equation \(9^{3/2} = 27\), where the square root of 9 is needed.
  • After finding the square root, the next task is usually to deal with any remaining exponent.
Knowing how to compute square roots is essential since it simplifies the process of working with powers and roots in various math problems.
When you see a fractional exponent, remember that the numerator is the power, and the denominator signifies the root. Hence, \(9^{3/2}\) tells us both to take a square root and then a cube.
Cube Numbers
A cube number is a number multiplied by itself twice. Mathematically, this is expressed as \(n^3\). For example:
  • The cube of 3 is \(3 \times 3 \times 3\), which equals 27.
  • Cubing a number involves three identical factors.
  • Cube numbers can quickly increase as you deal with larger base numbers.

In our problem, once we took the square root of 9 to get 3, the next step was to cube it. This resulted in \(3^3 = 27\), matching our target number perfectly.
Cube numbers are vital to understand, as they commonly appear in problems involving volume calculations and equations with higher exponents.
Equation Verification
Equation verification is the process of confirming that both sides of an equation are equal after performing mathematical operations.In any problem involving equations, it is crucial to check if our solution satisfies the original condition. Let's explore this further.
  • Verification involves substituting the calculated value back to the original equation.
  • It ensures no errors occurred during mathematical operations.

Looking at our step-by-step solution \(27 = 27\), there is a clear indication that the left-hand side equals the right-hand side.
This confirms the solution is correct.Verification is not just a step; it acts as a final check, enhancing confidence in the solution's accuracy and ensuring all calculations fulfill the equation's original requirements.

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Most popular questions from this chapter

\(h(x)=\ln (x+1)\)

\(x=e^{-2 / 3}\)

In Exercises 9-16, write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3\) \(5^{3}=125\)

\(y=\log _{5}(x-1)+4\)

\(\log _{8} 4=\frac{2}{3}\)
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