Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression with positive exponents and simplify.\((y+2)^{-2}(y+2)^{-1}\)

Step-by-Step Solution

Answer

The expression \((y+2)^{-2}(y+2)^{-1}\) when rewritten with positive exponents and simplified gives \(\frac{1}{(y+2)^3}\).

Step 1: Understanding the property of exponents

When multiplying like bases, the exponents should be added. So, \((y+2)^{-2} \times (y+2)^{-1} = (y+2)^{-2+(-1)}\).

Step 2: Adding the exponents

After adding the exponents, -2 and -1, \((y+2)^{-2+(-1)} = (y+2)^{-3}\).

Step 3: Rewriting the exponent to a positive value

A negative exponent in the numerator is the same as a positive exponent in the denominator and vice versa. Therefore, the expression \((y+2)^{-3}\) can be rewritten as \(\frac{1}{(y+2)^3}\) to put the exponent into positive form.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, when we see something like \(a^n\), it's telling us to multiply \(a\) by itself \(n\) times. It's a shorthand that makes working with large numbers of multiplications much simpler.

• For example, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
• This handy notation is especially useful in algebraic expressions where variables may be involved.

When multiplying expressions with the same base, you add their exponents. This is because you're essentially extending the chain of multiplications. Consider \(x^m \times x^n = x^{m+n}\), which shows how the rule functions practically by just adding the exponents \(m\) and \(n\).
Understanding exponents is fundamental when simplifying expressions and solving algebraic equations.

Simplifying Expressions

Simplifying an algebraic expression means rewriting it in a simpler or more efficient form without changing its value. The goal is often to make the expression easier to work with or understand. This involves combining like terms, applying any applicable rules of arithmetic, and in this case, managing the exponents.
For our specific exercise, simplifying means combining the exponents of the expressions that have the same base.

• Start with associating the base, like \((y+2)\) in our example.
• Then, use the rule of adding exponents when multiplying terms with the same base.
• Finally, simplify further if possible, converting negative exponents into positive ones.

By following these steps, complex algebraic expressions become more manageable and provide a clear path to the solution.

Negative Exponents

Negative exponents represent reciprocals. They are a way to express the inverse of a number raised to a power. For instance, \(a^{-n} = \frac{1}{a^n}\), which shows the principle that a negative exponent 'flips' the base to the denominator.
This is crucial to understand when simplifying expressions. In the original exercise, the negative exponent \((y+2)^{-3}\) is converted to \(\frac{1}{(y+2)^3}\). This conversion is based on the rule that negative exponents indicate division rather than multiplication.

• Remember, a negative exponent means "take the reciprocal".
• This not only simplifies expressions but also makes them mathematically correct by ensuring all exponents are positive by the end of simplification.

Using these concepts effectively will help in avoiding errors and arriving at simplified, more understandable expressions.