Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression with positive exponents and simplify.\(\left(\frac{y}{5}\right)^{-2}\)

Step-by-Step Solution

Answer

The expression \(\left(\frac{y}{5}\right)^{-2}\) rewritten with positive exponents and simplified is \(25/y^2\).

Step 1: Understand the Exponent Laws

An expression with a negative exponent is equivalent to its reciprocal with a positive exponent. This is because \(b^{-n} = 1/b^n\) for any real number \(b\) and positive integer \(n\). Therefore, we can rewrite \(\left(\frac{y}{5}\right)^{-2}\) as \(\left(\frac{5}{y}\right)^2\).

Step 2: Simplify

We can now proceed and simplify the expression. Remember, when squaring a fraction, we square both the numerator and the denominator. Squaring the numerator \(5^2\) gives 25, and squaring the denominator \(y^2\) gives \(y^2\). So the simplified expression now becomes \(25/y^2\).

Key Concepts

Exponent Laws

When dealing with exponents, especially negative exponents, it's crucial to understand the basic laws. Exponent laws are rules that help us simplify expressions that involve powers. Knowing these laws makes it easier to manipulate and solve expressions.

• One key rule is that any base raised to a negative exponent is equivalent to one divided by that base raised to the opposite positive exponent. Mathematically, this law is expressed as:\[ b^{-n} = \frac{1}{b^n} \]
• Understanding this helps in transforming expressions with negative exponents to those with positive exponents, which are generally easier to work with.
• In our example, we used this rule to convert \(\left(\frac{y}{5}\right)^{-2}\) into \(\left(\frac{5}{y}\right)^2\).

Becoming familiar with exponent laws provides a strong foundation for working with exponential expressions and reliably simplifies complex mathematical problems.

Reciprocal

The concept of a reciprocal is closely tied to negative exponents. A reciprocal of a number is simply the result you get when you divide 1 by that number. So, if you have a fraction like \(\frac{a}{b}\), its reciprocal would be \(\frac{b}{a}\).

• In terms of exponents, when you encounter a negative exponent, taking the reciprocal is one of the initial steps to simplify the expression.
• For example, with the expression \(\left(\frac{y}{5}\right)^{-2}\), the negative exponent suggests that we need to take the reciprocal of the fraction first.
• This results in flipping the fraction to get \(\left(\frac{5}{y}\right)^2\), effectively changing the exponent from negative to positive.

By understanding reciprocals, you can easily handle similar expressions and immediately convert them into a simpler positive exponent form.

Simplifying Expressions

Simplifying expressions involves using mathematical operations to reduce them to their simplest form. This often includes removing unnecessary parts or combining like terms.

• After converting an expression to positive exponents, the next step is to simplify it as much as possible.
• For fractions raised to a power, like \(\left(\frac{5}{y}\right)^2\), you should square both the numerator and the denominator separately.
• Squaring the fraction gives us \(\frac{5^2}{y^2}\), which simplifies to \(\frac{25}{y^2}\).
• This final expression is much easier to interpret and manage in subsequent calculations.

With practice, simplifying expressions becomes second nature, allowing you to focus on more complex problem-solving aspects.