Problem 70 - Algebra 1

Question

Simplify the expression. Then evaluate the expression when \(a=1\) and \(b=2\). $$\left(a^{3} b\right)^{4}$$

Step-by-Step Solution

Answer

16

Step 1: Apply Power of a Product Rule

Use the rule of exponents \((ab)^n = a^n * b^n\) to simplify \(\left(a^{3}b\right)^{4} \)to \(a^{12} b^{4}\).

Step 2: Substitute Values for a and b

Substitute \(a = 1\) and \(b = 2\) into the simplified expression \(a^{12} b^{4}\) which gives us \(1^{12}*2^{4}\).

Step 3: Simplify the Expression

Do the calculation to make the final simplification which gives us \(1*16 = 16\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. When you see an exponent, it indicates how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(a^3\), the base \(a\) is multiplied by itself three times, or \(a \times a \times a\). Exponents allow for simplified representation of large numbers and complex equations.
Exponents also have rules that make working with them easier. One such rule is multiplying powers with the same base. For example, \(a^m \times a^n = a^{m+n}\). This rule helps simplify expressions that might otherwise be difficult to work with. Remembering these rules will help you tackle various problems involving exponents.

Power of a Product

The "Power of a Product" rule is an important exponent rule that simplifies expressions where both a product and a power are involved. According to this rule, when you have an expression like \((ab)^n\), you can distribute the exponent \(n\) to both the factors inside the parenthesis: \((ab)^n = a^n \cdot b^n\).
This rule is particularly helpful when simplifying expressions like \((a^3 b)^4\). Here, you apply the power of 4 both to \(a^3\) and to \(b\). This transforms the expression into \(a^{12} \cdot b^4\), which simplifies future calculations. Understanding and applying the "Power of a Product" rule can make handling expressions with exponents much more straightforward.

Substitution Method

The substitution method involves replacing variables with their numerical values to evaluate an algebraic expression. After simplification with rules like the power of a product, substitution helps you calculate actual values.
In our simplified expression \(a^{12}b^4\), given \(a = 1\) and \(b = 2\), substitution allows us to replace \(a\) and \(b\) with these values, yielding \(1^{12} \cdot 2^4\). Since \(1^{12}\) equals 1, and \(2^4\) equals 16 (calculated as \(2 \times 2 \times 2 \times 2\)), substituting these values helps us find that \(1 \cdot 16 = 16\).
This method is essential in solving real-world math problems and converting variables into real numbers.