Problem 64 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(\frac{4^{2} a^{3} b^{7}}{b^{5} c^{4}}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\left(\frac{(4^{2})^{2} (a^{3})^{2} (b^{7})^{2}}{(b^{5})^{2} (c^{4})^{2}}\right)$$ Answer: The simplified expression is: $$\frac{256a^6b^4}{c^8}$$

Step 1: Apply power-of-a-power rule

Start by applying the power-of-a-power rule to the expression inside the brackets. This rule states that to raise a product or quotient to a power, raise both the numerator and the denominator to that power. So the given expression becomes: $$ \left(\frac{(4^{2})^{2} (a^{3})^{2} (b^{7})^{2}}{(b^{5})^{2} (c^{4})^{2}}\right) $$

Step 2: Simplify the exponents

Now, simplify the exponents for each term. Using the rule \((a^m)^n = a^{mn}\), the expression becomes: $$ \frac{4^{4} a^{6} b^{14}}{b^{10} c^{8}} $$

Step 3: Combine like terms

Combine the terms with the same base to make the expression simpler. For this, we use the quotient of powers rule, which states that \(\frac{a^m}{a^n} = a^{m-n}\), so the given expression becomes: $$ \frac{4^{4} a^{6} b^{14-10}}{c^{8}} $$

Step 4: Simplify the expression

Finally, calculate the result of the simplified exponents: $$ \frac{4^{4} a^{6} b^{4}}{c^{8}} $$ Thus, the simplified expression is: $$ \frac{256a^6b^4}{c^8} $$

Key Concepts

Power Rules for Exponents

Understanding the power rules for exponents is essential for simplifying mathematical expressions. Exponents, also called powers, indicate the number of times a base is multiplied by itself. The power rules are a set of guidelines that help us manipulate expressions involving exponents more efficiently.

One of the fundamental power rules is the multiplication of powers with the same base, which states that when multiplying exponents with the same base, you add the exponents: \( a^m \cdot a^n = a^{m+n} \). This rule significantly shortens the process of multiplying large numbers.

Another rule is the power of a product rule. When you have a product raised to an exponent, each factor of the product is raised to that exponent: \( (ab)^n = a^n \cdot b^n \). Similarly, the power of a quotient rule says that for a quotient raised to an exponent, both the numerator and the denominator are raised to that exponent: \( (\frac{a}{b})^n = \frac{a^n}{b^n} \).

These power rules are not just theoretical; they are practical tools that, once mastered, can make complex algebraic operations simpler and more manageable. Regular practice with these rules will make them a natural part of your mathematical skill set.

Power-of-a-Power Rule

The power-of-a-power rule is a powerful tool in algebra that simplifies expressions where an exponent is raised to another exponent. According to this rule, when you have an exponent raised to another exponent, you multiply the exponents. The mathematical representation is \( (a^m)^n = a^{mn} \).

Let's demonstrate this with an example from the exercise. When we simplify \( (4^2)^2 \), we multiply the exponents: \( 4^{2 \cdot 2} = 4^4 \). We apply the same principle to \( (a^3)^2 \) and \( (b^7)^2 \) to obtain \( a^6 \) and \( b^{14} \) respectively.

The power-of-a-power rule streamlines calculations involving exponential terms. Memorizing this rule, along with practicing its application, will yield quick and accurate results in mathematical problems involving powers.

Quotient of Powers Rule

When simplifying exponents, the quotient of powers rule is a shortcut for dividing terms with the same base. This rule states that when you divide exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).

Following this rule, in the exercise, we are presented with \( b^{14} \div b^{10} \). Applying the quotient of powers rule gives us \( b^{14-10} \) or \( b^4 \). This simplification highlights how exponents can be easily managed when dividing same-base terms.

It’s important to remember that the quotient of powers rule can only be used when the bases are identical. This rule, combined with the others, forms a toolkit that can be used to tackle a wide array of problems involving exponents. Understanding and implementing the quotient of powers rule facilitates significant simplification in algebra and higher mathematics.