Problem 52 - Intermediate Algebra

Question

Simplify each expression. Write answers using positive exponents. $$ \frac{y^{-3} y^{-4} y^{0}}{\left(2 y^{-2}\right)^{3}} $$

Step-by-Step Solution

Answer

\( \frac{1}{8y} \)

Step 1: Apply Exponent Rules to the Numerator

In the numerator, we have terms with the same base 'y' that are being multiplied together: \( y^{-3} y^{-4} y^{0} \). Using the rule \( a^m a^n = a^{m+n} \), we combine the exponents: \( y^{-3-4+0} = y^{-7} \).

Step 2: Expand the Denominator by Applying Power Rule

In the denominator, \( \left(2y^{-2}\right)^3 \) means we need to apply the power rule, which is \( (ab)^n = a^n b^n \). Thus we have \( 2^3 (y^{-2})^3 = 8 y^{-6} \).

Step 3: Simplify the Complete Expression

We now have \( \frac{y^{-7}}{8y^{-6}} \). Using the formula \( \frac{a^m}{a^n} = a^{m-n} \), this becomes \( \frac{y^{-7}}{y^{-6}} = y^{-7 - (-6)} = y^{-1} \).

Step 4: Finalize the Expression

Now substitute back into the expression we handled above: \( \frac{y^{-1}}{8} = \frac{1}{8y} \). The numerator was simplified completely and turned into \( y^{-1} = \frac{1}{y} \), resulting in our final expression with positive exponents.

Key Concepts

Negative Exponents

When working with negative exponents, it's crucial to understand their meaning. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.
For example, if you have an expression like \( y^{-3} \), this is equivalent to \( \frac{1}{y^3} \).
Essentially, it tells you to "flip" the base to the other side of a fraction.

• Every time you encounter a negative exponent, convert it by flipping the base and changing the sign to positive.
• This process helps in rewriting expressions in a form that's easier to understand and manipulate, especially when multiplying or dividing.
• Always aim to have positive exponents in your final answer to adhere to standard conventions.

Simplifying Expressions

Simplifying expressions is a key component in algebra and involves condensing complex expressions into simpler forms. Here's how you can approach it.

• Look to combine like terms, especially those with the same base such as \(y^{-3} y^{-4} y^{0}\).
• The exponent rules, especially multiplication and division rules, will be your tools.
• Apply the rule \(a^m a^n = a^{m+n}\) to simplify bases with multiple exponents.
• For division, use \(\frac{a^m}{a^n} = a^{m-n}\) to resolve exponents across division bars.

After using these rules, you aim to express the given term with the simplest positive exponents possible, making it easier to interpret and solve any forthcoming algebraic tasks.

Power Rule

The power rule focuses on simplifying expressions where an exponent is raised to another power.
This is beautifully summarized by \((ab)^n = a^n b^n\), letting us "distribute" the outer exponent to each component inside the parenthesis.

• In our context, consider the expression \(\left(2y^{-2}\right)^3\)
• Apply the power rule by making it \(2^3 (y^{-2})^3\), expanding each factor.
• Calculate \(2^3\), which yields 8, and multiply the exponents on the \(y\), giving \(y^{-6}\).

This rule is particularly powerful for breaking down compounded expressions, simplifying them into components that can then be further reduced or managed according to other exponent rules. Always separate the coefficients and bases to simplify efficiently.