Problem 32 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ (x+5)^{-2} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((x+5)^{-2}\) with a positive exponent. Answer: \(\frac{1}{(x+5)^2}\)

Step 1: Recall the rule for negative exponents

To rewrite the given expression with positive exponents, remember that a number or variable raised to a negative exponent can be replaced by its reciprocal raised to the positive exponent. More formally: $$ a^{-n} = \frac{1}{a^n} $$

Step 2: Apply the rule to the given expression

Now, apply the rule mentioned in step 1 to the given expression \((x+5)^{-2}\): $$ (x+5)^{-2} = \frac{1}{(x+5)^2} $$ So, the expression \((x+5)^{-2}\) with a negative exponent is equivalent to \(\frac{1}{(x+5)^2}\) with a positive exponent.

Key Concepts

Negative Exponents

Understanding negative exponents can make algebraic expressions much easier to handle. A negative exponent indicates how many times we need to divide by the base number. Here’s a simple way to remember it:

• The expression \(a^{-n}\) implies dividing 1 by \(a^n\).
• It flips the base number to take its reciprocal.

For example, when you see \((x+5)^{-2}\), it is equivalent to saying "take one divided by \((x+5)^2\)". The rule for negative exponents can simplify calculations and is vital for working with fractions.

Reciprocal

The reciprocal is a concept closely tied to negative exponents. In mathematics, the reciprocal of a number \(a\) is simply \(\frac{1}{a}\). This term is essential when you want to turn negative exponents into positive ones.

• Changing \((x+5)^{-2}\) into \(\frac{1}{(x+5)^2}\) is a direct application of using reciprocals.
• Reciprocals are useful in simplifying complex fractions and algebraic expressions.

Remember, the reciprocal flips the number or expression, emphasizing the division aspect. This concept is useful in both algebra and calculus, as it turns multiplication into division, and vice versa.

Positive Exponents

Positive exponents are straightforward compared to negative ones. When you see a positive exponent, it tells you how many times to multiply the base by itself.

• For example, \(a^n\) means multiply the base \(a\) by itself \(n\) times.
• Positive exponents indicate repeated multiplication, making them easy to understand and apply in equations.

Using positive exponents is intuitive when simplifying expressions, such as changing \((x+5)^{-2}\) to \(\frac{1}{(x+5)^2}\). By focusing on positive exponents, algebraic problems become more manageable and neat.