Question

Evaluate the expression. \(\frac{45}{a^{2}}\) when \(a=2\)

Short answer

The value of the expression \(\frac{45}{a^{2}}\) when \(a=2\) is 11.25

Step-by-step solution

Substitute 'a' Value

We begin by replacing the 'a' in the expression with the value given, which is 2. This gives us the new expression: \(\frac{45}{2^{2}}\)

Square 'a' Value

We square the denominator as indicated by the exponent 2. This gives us \(\frac{45}{2*2} = \frac{45}{4}\)

Divide Numerator by Denominator

Next, we perform the division of 45 by 4 to get the final value of the expression: 11.25

Key concepts

Substitution in Algebra

Substitution is a fundamental skill in algebra that allows us to evaluate expressions by replacing variables with their given values. When you see an expression like \(\frac{45}{a^{2}}\) and you're told that \(a=2\), substitution tells us to plug in the value of \(a\) into the expression.

To carry out substitution effectively, you need to carefully replace every instance of the variable with its value. In our exercise, after substituting \(2\) for \(a\), the expression becomes \(\frac{45}{2^{2}}\). Always ensure you've substituted every occurrence of the variable and that no variable is left unchanged unless stated.

Why Is Substitution Important?

Substitution is key to solving equations and understanding how changes in one variable affect the outcome. It also helps in verifying solutions to problems and can serve as a stepping stone for more complex mathematical concepts like functions.

Exponents and Powers

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. The exponent is the small number written above and to the right of the base number, indicating the power to which the base is raised.

For instance, \(2^{2}\) means \(2\) is multiplied by itself once: \(2 * 2 = 4\). Understanding how to work with exponents is crucial because it simplifies expressions and calculations, particularly when dealing with large numbers.

Working with Exponents

When evaluating expressions with exponents, like the \(2^{2}\) in our exercise, always calculate the power before any other operation. This is in line with the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By applying the exponent first, we ensure that the calculations proceed correctly and that we reach the right answer.

Arithmetic Division

Division is one of the four basic arithmetic operations, alongside addition, subtraction, and multiplication. It involves splitting a number, known as the dividend, into a specified number of parts, which is the divisor, to find how many times the divisor fits into the dividend.

When dividing numbers, any remainders can be left as they are for a quotient that is a whole number, or they can be expressed as decimals or fractions. In our original problem, dividing \(45\) by \(4\) results in the decimal \(11.25\).

Tips for Division

To improve division skills, it’s important to be comfortable with basic times tables and to understand the relationship between division and multiplication. When a problem looks complex, breaking it down into smaller parts or using a calculator to check work can help ensure accuracy in the final result.