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Chapter 12: Problem 51

Evaluate each of the following. $$ 25^{1 / 2} $$

Short Answer

Expert verified
The value of \(25^{1/2}\) is 5.

Step by step solution

01

Understand the Problem

We need to evaluate the expression \( 25^{1/2} \). This requires finding the value of 25 raised to the power of \( \frac{1}{2} \).
02

Interpret the Exponent

The exponent \( \frac{1}{2} \) indicates a square root. Therefore, \( 25^{1/2} \) is equivalent to the square root of 25.
03

Calculate the Square Root

Identify the number that when multiplied by itself gives 25. The number 5 satisfies \( 5 \times 5 = 25 \). Thus, \( \sqrt{25} = 5 \).
04

Conclude the Evaluation

Therefore, \( 25^{1/2} = 5 \).

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Key Concepts

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

Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, the square root of 25 is written as \( \sqrt{25} \).

For example:
  • The square root of 16 is 4, because \( 4 \times 4 = 16 \).
  • The square root of 9 is 3, because \( 3 \times 3 = 9 \).
  • Similarly, the square root of 25 is 5, because \( 5 \times 5 = 25 \).
Square roots are very common in mathematics, and they often appear when dealing with quadratic equations and geometry. Understanding that finding a square root is essentially "reversing" the process of squaring a number will help you evaluate these expressions effectively.
Evaluating Expressions
Evaluating a mathematical expression means calculating its value. This involves replacing variables and constants with numbers, or completing operations as specified by the expression.

Take the expression \( 25^{1/2} \) for instance:
  • This expression means that we need to find the square root of 25, as explained before.
  • In this case, the evaluation involves recognizing that the exponent \( \frac{1}{2} \) refers to finding a square root.
  • The calculation tells us that the expression evaluates to 5.
When evaluating expressions, it's crucial to follow a systematic approach:
  • Identify what each component of the expression signifies (such as exponents).
  • Carry out the necessary operations in the correct order (like finding roots or powers).
Evaluating expressions accurately requires understanding the operations involved and often involves breaking down complex expressions into simpler parts.
Rational Exponents
Rational exponents are exponents that are fractions, such as \( \frac{1}{2}, \frac{1}{3}, \frac{3}{4} \), and so on. These exponents indicate roots of numbers rather than traditional powers.

With rational exponents, the numerator usually denotes the power, and the denominator indicates the root. For instance:
  • \( a^{1/2} \) is equivalent to the square root of \( a \), written as \( \sqrt{a} \).
  • \( a^{1/3} \) signifies the cube root of \( a \), or \( \sqrt[3]{a} \).
  • \( a^{m/n} \) means that you take the \( n^{th} \) root of \( a \) and then raise the result to the \( m^{th} \) power.
Understanding rational exponents is essential because they provide a concise way to denote roots and powers within equations. Transitioning smoothly between fractional exponents and roots can greatly simplify complex algebraic manipulations. Using rational exponents allows for more flexibility and efficiency in solving mathematical problems.

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Most popular questions from this chapter

Graph \(f(x)=\ln x(\text { Exercise } 89), f(x)=\ln x-3 \text { (Exercise } 95)\) and \(f(x)=\ln x+3(\text { Exercise } 96) .\) Discuss any trends shown on the graphs.

Find the inverse of each given one-to-one function. Then use a graphing calculator to graph the function and its inverse on a square window. $$ f(x)=\sqrt[3]{x+1} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=-2 e^{x} $$

Use the formula \(A=P e^{r t}\) to solve. See Example 8. Find how much money Dana Jones has after 12 years if 1400 dollars is invested at \(8 \%\) interest compounded continuously.

Solve each equation for \(x\). Give an exact solution and a four-decimal-place approximation. See Examples 3 and 7. $$ \ln (3 x-4)=2.3 $$
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