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Chapter 1: Problem 129

Evaluate. $$ \frac{x^{2}+2^{x}}{x^{2}-2^{x}}, \text { for } x=3 $$

Short Answer

Expert verified
The value of the expression is 17.

Step by step solution

01

- Substitute x with 3

To evaluate the expression \(\frac{x^{2}+2^{x}}{x^{2}-2^{x}}\) for \(x=3\), first substitute \(x\) with 3 in both the numerator and the denominator.
02

- Simplify the numerator

After substitution, the numerator becomes \(3^{2} + 2^{3}\). Calculate these values:\(3^{2} = 9\)\(2^{3} = 8\)So, the numerator is \(9 + 8 = 17\).
03

- Simplify the denominator

After substitution, the denominator becomes \(3^{2} - 2^{3}\). Calculate these values:\(3^{2} = 9\)\(2^{3} = 8\)So, the denominator is \(9 - 8 = 1\).
04

- Divide the simplified numerator by the simplified denominator

Finally, divide the simplified numerator, which is 17, by the simplified denominator, which is 1:\( \frac{17}{1} = 17 \).

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Key Concepts

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

Substitution Method
The substitution method is a fundamental concept in algebra. It involves replacing a variable with a given value. In this exercise, we are asked to evaluate the expression \( \frac{x^{2}+2^{x}}{x^{2}-2^{x}} \) for \( x=3 \).

To use the substitution method:
  • First, you identify the variable in the expression.
  • Next, replace this variable with the provided value.

Here, replacing \( x \) with 3 transforms the expression into \( \frac{3^{2} + 2^{3}}{3^{2} - 2^{3}} \). This substitution simplifies the evaluation process.
Simplifying Expressions
Simplifying expressions is essential for solving algebraic problems efficiently. In this exercise, after substituting \( x \) with 3, we need to simplify both the numerator and the denominator separately.

For the numerator:
  • Calculate \( 3^{2} = 9 \).
  • Next, calculate \( 2^{3} = 8 \).
  • Sum these values to get \( 9 + 8 = 17 \).

For the denominator:
  • Calculate \( 3^{2} = 9 \).
  • Next, subtract \( 2^{3} = 8 \) from 9.
  • This results in \( 9 - 8 = 1 \).

By simplifying, we make it easier to perform the final division.
Division in Algebra
Division in algebra involves dividing one expression by another. After simplifying the numerator and the denominator, the original expression becomes \( \frac{17}{1} \).

Performing this division is straightforward since any number divided by 1 remains unchanged. Therefore:
  • The result is \( 17 \).

Understanding this concept helps in tackling various algebraic expressions more confidently and correctly.

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Most popular questions from this chapter

Evaluate using a calculator. $$ 3(m+2 n) \div m, \text { for } m=1.6 \text { and } n=5.9 $$

Simplify. $$ 8 x-(6 x+7) $$

Evaluate. $$ \frac{x^{2}+2^{x}}{x^{2}-2^{x}}, \text { for } x=2 $$

Write an equivalent expression without using grouping symbols. $$ -(9 x+1) $$

Simplify. Match the algebraic expression with the equivalent rewritten expression below. Check your answer by calculating the expression by hand and by using a calculator. A) \((5(3-7)+4 \wedge 3) /(-2-3)^{2}\) B) \((5(3-7)+4 \wedge 3) /\left(-2-3^{2}\right)\) C)\((5(3-7)+4) \wedge 3 /-2-3^{2}\) D) \(5(3-7)+4 \wedge 3 /(-2-3)^{2}\) $$ \frac{5(3-7)+4^{3}}{(-2-3)^{2}} $$
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