Problem 60 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The value of \(\left(a^{2} b\right)^{4}\) when \(a=1\) and \(b=2\) is 16

Step 1: Apply the Power of Power Rule

Apply the power of power rule. Given \(\left(a^{2} b\right)^{4}\), the rule implies \((a^{2*4})*(b^{4}) = a^{8}b^{4}\)

Step 2: Substitute the Given Values

Substitute the given values \(a=1\) and \(b=2\) into the equation. This gives \(a^{8}b^{4} = 1^{8} * 2^{4}\)

Step 3: Evaluate the Expression

Evaluate the expression. Hence, \(1^{8} * 2^{4} = 1 * 16 = 16\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers, known as the base and the exponent. The base is the number being multiplied by itself, and the exponent indicates how many times this multiplication occurs. For example, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. Typically, exponentiation is written as \(a^n\) and pronounced as 'a raised to the power of n'.

• When the exponent is 1, the result is simply the base itself, i.e., \(a^1 = a\).
• If the exponent is 0, and the base is non-zero, the result is always 1, i.e., \(a^0 = 1\).
• When exponentiating a product, each component of the product is raised to the exponent, which is demonstrated by the power of a power rule.

This power of a power rule states that \( (a^m)^n = a^{m \times n} \). In the given problem, applying this rule to \( (a^2 b)^4 \) results in \( a^{2\times4} \cdot b^4, \) simplifying to \( a^8 \cdot b^4 \). Understanding how to manipulate exponents using their foundational rules is essential when simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. These expressions represent value relationships and are crucial for solving equations. Variables in these expressions often symbolize unknowns or quantities that can change. In \((a^2b)^4\), '\(a\)' and '\(b\)' are variables, while coefficients are constants that multiply the variable parts.
An essential skill is simplifying these expressions, often using laws of exponents and arithmetic. Once simplified, they can be evaluated by substituting values for the variables. For the expression \((a^2b)^4\), simplification involves applying the power of a power rule to break it down into a more straightforward form: \(a^8b^4\).

• Recognizing like terms is key to simplification, which requires combining coefficients and powers of variables correctly.
• The goal is to reduce the expression to its simplest form, optimizing for substitution.

Algebraic expressions allow for the precise modeling of real-world scenarios, providing solutions to practical problems.

Substitution Method

The substitution method is a straightforward procedure where numerical values for variables are substituted into an algebraic expression to find a numerical solution. By directly replacing variables with their given or calculated values, the process simplifies complex expressions into manageable calculations.
In the exercise with \(a = 1\) and \(b = 2\), substituting these values into the previously simplified expression \(a^8b^4\) turns it into \(1^8 \times 2^4\).

• Substitute every instance of the variable with its corresponding value.
• Carry out the arithmetic calculations to simplify into a single numerical result.

This method ensures clarity and accuracy when calculating the expression's value, leading to the final answer of 16. Substitution is especially useful in checking solutions and solving real-value problems where constants are known.