Problem 77 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(\frac{1}{y^{2}}\) when \(y=5\)

Step-by-Step Solution

Answer

The simplified expression for the given variable value is \(\frac{1}{25}\).

Step 1: Substitution

Start by substituting the given value for the variable \(y\) in the given expression. The given value of \(y=5\), substitute this in \(\frac{1}{y^{2}}\) to get \(\frac{1}{5^{2}}\).

Step 2: Exponentiation

Next, simplify \(5^{2}\) to \(25\). So, the expression becomes \(\frac{1}{25}\).

Step 3: Final expression

The simplified expression is \(\frac{1}{25}\) and this is as simplified as the expression can get, as there isn't another proper fraction that could represent this decimal.

Key Concepts

Substitution

Substitution is a key concept in algebra, where we replace a variable with its actual value. For example, in the expression \( \frac{1}{y^{2}} \), the variable is \( y \). This expression evaluates to a number once we know the value of \( y \). When \( y = 5 \), we substitute 5 in place of \( y \).

So, instead of dealing with \( y \), we now work with a specific number, making the expression more concrete: \( \frac{1}{y^{2}} \) becomes \( \frac{1}{5^{2}} \). This is a common technique used in mathematical problems and helps in solving equations.

Remember:

• Identify the variable in the expression.
• Replace it with the given value.

By applying substitution correctly, we make the expression ready for further simplification.

Exponentiation

Exponentiation is the next step in evaluating expressions like \( \frac{1}{5^{2}} \). Here, we deal with powers, indicated by the exponent. The exponent tells us how many times to multiply the base by itself. In our example, \( 5^{2} \) means \( 5 \times 5 \).

Calculating the exponentiation, we find that \( 5^{2} = 25 \). The expression has now turned from \( \frac{1}{5^{2}} \) to \( \frac{1}{25} \) by simply evaluating the power.

Key points:

• An exponent shows repeated multiplication of the base.
• Simplifying the power reduces the complexity of the expression.

Understanding exponentiation helps in transforming expressions from complex to simple, especially when preparing for the final simplification.

Simplification

Simplification is the process of breaking down expressions to their simplest form. After substitution and exponentiation, we had the expression \( \frac{1}{25} \). This represents a fraction that can't be reduced further. Thus, the process of simplification is complete here.

In mathematics, simplification means:

• Reducing fractions to the lowest terms.
• Making expressions easy to read and understand.
• Eliminating any unnecessary parts of the expression.

With the expression \( \frac{1}{25} \), we've reached a form that's fully simplified. This means there are no common factors in the numerator and denominator except 1. Understanding simplification helps in finishing the evaluation of expressions efficiently.