Problem 68 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable.(Lesson \(1.2)\). $$ (7 h)^{3} \text { when } h=1 $$

Step-by-Step Solution

Answer

The evaluated expression results in a value of 343.

Step 1: Substitute the Value

Replace the value of \(h\) in the expression. So, \(7h\) becomes \(7*1\).

Step 2: Simplify the Base

First, perform the multiplication operation to find the base of the power. This gives us a base of 7.

Step 3: Evaluate the Power

Next, calculate the result of \(7^3\). This results in a value of 343.

Key Concepts

Substitution Method

When tackling algebraic expressions, the substitution method is a powerful tool. It involves replacing a variable with a given value. This means if you have an expression like \((7h)^3\) and you know that \(h = 1\), you simply replace \(h\) with 1. Here’s how you do it step by step:

• Identify the variable and the value it equals, which in this case is \(h = 1\).
• Plug this value into the expression wherever the variable appears.

This substitution transforms the expression from \(7h\) to \(7 \times 1\). With substitution, you turn algebraic expressions into more straightforward arithmetic problems that are often much easier to solve.

Exponents in Algebra

Exponents signify how many times you multiply a number by itself. In algebra, you often deal with expressions featuring exponents, such as \(7^3\), which means 7 is multiplied by itself three times. To help clarify:

• Here, the base is 7 and the exponent is 3.
• The calculation is thus \(7 \times 7 \times 7\).

The result of this operation gives us \(343\). Exponents are a key concept in mathematics as they allow you to express repeated multiplication compactly, making complex calculations more manageable.

Arithmetic Operations

Arithmetic operations form the foundation of math. They include operations like addition, subtraction, multiplication, and division. Focusing on multiplication, as used in this exercise, let’s break it down further:

• Multiplication is a shortcut for repeated addition; for instance, \(7 \times 1\) is simply \(7\) because you're adding 7 zero times to itself.
• It is crucial as it transforms numbers and simplifies expressions particularly when combined with exponents.

When you multiply within an algebraic expression, as done with \((7 \times 1)^3\), you're simplifying parts so that evaluating powers, like \(7^3\), becomes seamless. Mastery of arithmetic operations is essential for manipulating and simplifying algebraic expressions effectively.