Question

Evaluate the expression for the given value of the variable. $$8 n \text { when } n=\frac{3}{16}$$

Short answer

The simplified expression is \(\frac{3}{2}\)

Step-by-step solution

Substitute the Given Value for \(n\)

Substitute \(\frac{3}{16}\) for \(n\) in the expression. This gives: \(8 * \frac{3}{16}\)

Simplify the Expression

Multiply 8 by \(\frac{3}{16}\) to simplify the expression. This equals \(\frac{24}{16}\)

Simplify the Fraction

Reduce \(\frac{24}{16}\) to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 8. So, the value will be \(\frac{3}{2}\)

Key concepts

Substitution in Algebra

Substitution is a fundamental technique in algebra where we replace variables with their given values. To correctly evaluate expressions using substitution, follow these steps:

• Identify the variable in the expression and its corresponding given value.
• Replace the variable with its given value, being careful to maintain any operations attached to the variable.
• If there are fractions, decimals, or parentheses, pay special attention to the order of operations to avoid errors.

For example, if we are given the expression \(8n\) and told that \(n=\frac{3}{16}\), we substitute \(\frac{3}{16}\) for \(n\), which transforms the expression to \(8 \times \frac{3}{16}\). Substitution is a crucial first step in solving algebraic expressions and is widely applicable in different areas of mathematics.

Simplifying Expressions

Simplification in algebra involves reducing expressions to their most basic form while maintaining their original value. This process often includes combining like terms, using the distributive property, and reducing fractions.

Combining Like Terms

Look for terms that share the same variable and exponent, and combine them by adding or subtracting their coefficients.

Distributive Property

Apply the distributive property if necessary: \(a(b + c) = ab + ac\).

Reducing Fractions

When the expression involves fractions, reduce them by finding the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Using our expression \(8 \times \frac{3}{16}\), simplifying it translates to multiplying 8 by each term of the fraction, resulting in \(\frac{24}{16}\). This expression can still be simplified further, leading to our next concept: working with fractions in algebra.

Fractions in Algebra

Managing fractions in algebra is often considered challenging, but understanding a few key principles can make it straightforward. Here’s how to approach expressions with fractions:

• Finding a Common Denominator: When adding or subtracting fractions, ensure they have a common denominator before combining them.
• Multiplying and Dividing: Multiply fractions by multiplying their numerators and denominators respectively. When dividing, flip the second fraction and then multiply.
• Simplifying Fractions: Always simplify fractions to their lowest terms by dividing the numerator and the denominator by their greatest common divisor (GCD).

In the solution to our given problem, \(\frac{24}{16}\) is not in its simplest form. This fraction can be simplified by dividing its numerator and denominator by the GCD, which is 8. Upon doing so, we achieve the simplified result of \(\frac{3}{2}\). Hence, knowing these principles allows one to handle fractions within algebraic expressions competently.