Problem 46 - 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. $$ \left(2 a^{2}\right)^{4}\left(3 a^{5}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left(2 a^{2}\right)^{4}\left(3 a^{5}\right)^{2}$. Answer: $144a^{18}$

Step 1: Apply power rule to all factors inside the parentheses

For each factor within the parentheses, we will raise the factor to the indicated power. This can be done using the rule \((ab)^n = a^n b^n\). In our case: \((2 a^{2})^{4} = 2^4 (a^{2})^4\) and \((3 a^{5})^{2} = 3^2 (a^{5})^2\).

Step 2: Simplify the expression

Now, simplify the expression by calculating the result of each exponent calculation: - \(2^4 = 16\) - \((a^{2})^4 = a^{2 \times 4} = a^8\) (using the rule \((a^m)^n = a^{m \times n}\)) - \(3^2 = 9\) - \((a^{5})^2 = a^{5 \times 2} = a^{10}\) Now our expression becomes: $$ 16 a^8 \cdot 9 a^{10} $$

Step 3: Combine terms with the same base

Lastly, combine the terms involving the variable \(a\). Since we're multiplying expressions with the same base, we can use the rule \(a^m \cdot a^n = a^{m + n}\). In our case, \(a^8 \cdot a^{10} = a^{8 + 10} = a^{18}\). Now the fully simplified expression is: $$ 16 \cdot 9 a^{18} $$

Step 4: Multiply the constants

Finally, multiply the constants 16 and 9 to simplify the expression even further: $$ 16 \cdot 9 = 144 $$ The final simplified form of the given expression is: $$ 144 a^{18} $$

Key Concepts

Power Rules of Exponents

When working with exponents, understanding the power rules is crucial. These rules allow you to simplify expressions where numbers or variables are raised to powers, making it easier to handle complex calculations.
\[ (a^m)^n = a^{m \times n} \] is a key rule called the "Power of a Power". This rule states that when raising an exponent to another exponent, you multiply the exponents.
For instance:

• \( (2^2)^3 = 2^{2 \times 3} = 2^6 \)
• \( (a^4)^5 = a^{4 \times 5} = a^{20} \)

Another important rule is the "Power of a Product", which tells us how to handle products within parentheses:
\[ (ab)^n = a^n \times b^n \] applies to any number or variable raised to a power and multiplied. This means that both the base and the variable should be raised to the specified power inside the parentheses.

• \( (3a)^2 = 3^2 \cdot a^2 \)
• \( (xy)^3 = x^3 \cdot y^3 \)

Knowing these power rules empowers you to break down and simplify complex exponent expressions efficiently.

Simplifying Expressions Using Exponents

Simplifying expressions with exponents helps you reduce lengthy combinations into simpler, more manageable forms. This process usually involves applying power rules effectively.
When given an expression like \( (2a^2)^4(3a^5)^2 \), you start by applying the power rules to simplify each part:

• Perform calculations using the Power of a Product: \( (2a^2)^4 = 2^4 \cdot (a^2)^4 \)
• Multiply each exponent where necessary: \( (a^2)^4 = a^{2 \times 4} = a^8 \)
• Next, do the same for the second part: \( (3a^5)^2 = 3^2 \cdot (a^5)^2 \)
• And the multiplication gives: \( (a^5)^2 = a^{5 \times 2} = a^{10} \)

Then, combine terms with the same base:

• \( a^8 \cdot a^{10} = a^{8+10} = a^{18} \)

Finally, multiply constants:

• \( 16 \cdot 9 = 144 \)

The expression then becomes \( 144a^{18} \), fully simplified. This systematic approach reduces lengthy expressions and prevents errors.

Understanding Natural Numbers in Exponents

Natural numbers are integers starting from 1 and increasing indefinitely (1, 2, 3, ...). They are essential in arithmetic and algebra, especially when dealing with exponents, as they typically denote how many times a number or a base variable is multiplied by itself.
When we work with exponents, we apply natural numbers to indicate repeated multiplication:

• \( a^2 \) means \( a \cdot a \)
• \( a^3 \) means \( a \cdot a \cdot a \)

Natural numbers make it straightforward to calculate exponential growth. These numbers are always positive because the concept focuses on multiplying a base, showing how repeated operations affect the size or length of a number when combined in expressions.
When every variable and constant in expressions like \( (2a^2)^4 \) uses natural numbers, you can easily manage calculations by using power rules, knowing that there will be no fractions or negative exponents, simplifying each step consistently.