Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression with positive exponents and simplify.\(\left(4 y^{-2}\right)\left(8 y^{4}\right)\)

Step-by-Step Solution

Answer

The expression \(\left(4 y^{-2}\right) \left(8 y^{4}\right)\) with positive exponents and simplified is \(32y^2\).

Step 1: Transform the negative exponent into a positive

A negative exponent can be turned into a positive exponent by flipping the base to the denominator of a fraction, so \(4y^{-2}\) becomes \(\frac{4}{y^2}\).

Step 2: Multiplication without brackets

Multiply the numbers outside the brackets with their corresponding pair, so \(\frac{4}{y^2}\) * \(8y^4\) becomes \(\frac{32y^4}{y^2}\).

Step 3: Simplify the expression

Simplify the expression by subtracting the exponents of 'y' in the numerator and the denominator. In this case, it becomes \(32(y^{4-2})=\) \(32y^2\).

Key Concepts

Negative Exponents

When dealing with algebra, negative exponents can initially seem confusing, but they are quite manageable once you grasp their meaning. A negative exponent indicates that the base should be moved to the opposite side of a fraction to become positive. For instance, if you encounter \(y^{-2}\), this means \(\frac{1}{y^2}\). In the context of our exercise, the expression \(4y^{-2}\) can be rewritten as \(\frac{4}{y^2}\).

• Negative exponents signify reciprocal action.
• To convert to a positive exponent, move the base to the other side of the fraction.
• This operation does not affect the base number itself, only its exponent.

Understanding this makes working with negative exponents straightforward, which is essential for simplifying and solving algebraic expressions.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing the expression to its simplest form. It often involves combining like terms and performing basic arithmetic using the laws of exponents. In the provided exercise, we combined and simplified terms such as \(y\) by following the exponent rules.To simplify the expression \(\frac{32y^4}{y^2}\):

• Apply the law of exponents: Subtract the exponent in the denominator from the exponent in the numerator.
• This results in \(y^{4-2} = y^2\).
• The simplified form becomes \(32y^2\), which is much cleaner.

Simplifying makes the expression easier to understand and work with, highlighting the core values you're working with.

Algebraic Multiplication

Algebraic multiplication involves multiplying coefficients and following rules of exponents for powers of variables. This process can also include working with expressions in parentheses. For the expression \((4y^{-2})(8y^4)\):

• First, multiply the coefficients \(4\) and \(8\) to get \(32\).
• For the variable terms, apply the laws of exponents separately by subtracting the exponents of like bases when dividing.
• In this case, \(y^{-2}\) and \(y^4\) becomes \(y^{4-2}\), simplifying to \(y^2\).

This reveals the combined result of direct multiplication, which gives us \(32y^2\). It's a powerful method to condense complicated expressions into more manageable and simplified forms.