Problem 70 - Elementary Algebra

Question

Write \(\left(\frac{4^{4} a^{8} b^{10}}{4^{2} a^{6} b^{2}}\right)^{-1}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Rewrite the given expression with only positive exponents: \(\left(\frac{4^{4} a^{8} b^{10}}{4^{2} a^{6} b^{2}}\right)^{-1}\) Answer: \(\frac{1}{16a^2b^8}\)

Step 1: Simplify the expression using the Quotient of Powers

Apply the quotient of powers rule to each term: \(\frac{a^m}{a^n} = a^{m-n}\). So, \(\frac{4^{4} a^{8} b^{10}}{4^{2} a^{6} b^{2}}=4^{(4-2)} a^{(8-6)} b^{(10-2)}\).

Step 2: Calculate the exponents

Calculate the new exponents: \(4^{4-2}=4^{2}\), \(a^{8-6}=a^{2}\), \(b^{10-2}=b^{8}\). Now we have \(\left(4^{2} a^2 b^8\right)^{-1}\).

Step 3: Apply the Power to Power Rule

The power to power rule states that \((a^{m})^n=a^{mn}\). Apply this rule to each term: \((4^{2})^{-1}=4^{-2}\), \((a^2)^{-1}=a^{-2}\), \((b^8)^{-1}=b^{-8}\). Now we have \(4^{-2} a^{-2} b^{-8}\)

Step 4: Rewrite with positive exponents

A negative exponent can be turned into a positive exponent by placing it in the denominator of a fraction: \(a^{-m} = \frac{1}{a^m}\). Thus, \(4^{-2} a^{-2} b^{-8} = \frac{1}{4^2 a^2 b^8} = \frac{1}{16a^2b^8}\). The final expression with only positive exponents is \(\frac{1}{16a^2b^8}\).

Key Concepts

Quotient of Powers

In mathematics, the quotient of powers is a fundamental rule that simplifies expressions involving division of exponents with the same base. When you divide like bases, you can subtract the exponents:

• If you have an expression like \( \frac{a^m}{a^n} \), you simplify it to \( a^{m-n} \).

This concept allows us to handle expressions by reducing them to more manageable forms. Let's use this rule in our original exercise:

\( \frac{4^{4} a^{8} b^{10}}{4^{2} a^{6} b^{2}} = 4^{(4-2)} a^{(8-6)} b^{(10-2)} \).

The exponents of \( 4 \), \( a \), and \( b \) are subtracted accordingly: \( 4^2 \), \( a^2 \), and \( b^8 \). This greatly simplifies our expression before moving on to further steps.

Negative Exponents

Negative exponents present a unique way to express reciprocal values in math. When you see a negative exponent, it simply means that you need to take the reciprocal of the base raised to the positive of that exponent:

• For \( a^{-m} \), it becomes \( \frac{1}{a^m} \).

This shift from negative to positive exponents ensures clarity in mathematical expressions by always representing exponents in a positive form. In the context of our exercise, after initially simplifying the expression using the quotient of powers, we're left with a negative exponent:

\( \left(4^{2} a^2 b^8\right)^{-1} \).

By applying the concept of negative exponents, we rewrite the expression as:

\( 4^{-2} a^{-2} b^{-8} \).

Later, these negative exponents become positive as we convert them to their reciprocal form.

Power to Power Rule

The power to power rule is a simple yet powerful tool for simplifying expressions involving exponents. This rule states that when you raise an exponent to another power, you multiply the exponents:

• If you have \((a^m)^n\), it becomes \(a^{m \cdot n}\).

So, how does this fit into simplifying expressions with exponents? After applying the quotient of powers and identifying negative exponents, our problem transforms:

We start with \((4^{2})^{-1}\), \((a^2)^{-1}\), and \((b^8)^{-1}\).

Applying the power to power rule, we find:

• \((4^{2})^{-1} = 4^{-2}\)
• \((a^{2})^{-1} = a^{-2}\)
• \((b^{8})^{-1} = b^{-8}\)

This step is crucial as it sets up the transition to expressing the final solution with only positive exponents.