# Evaluating Expressions

*Evaluating expressions* means replacing the variable with a specific value and then simplifying.

We can represent a general mathematical expression using letters, e.g. \(a+b = b+a\). To state the commutativity of real numbers under addition, we do not have to list all pairs of real numbers like

\[\begin{array} &1+2 = 2+1, &0.3 + \sqrt{2} = \sqrt{2} + 0.3, &\ldots. \end{array}\]

In the expression \(a+b = b+a,\) letters \(a\) and \(b\) are called variables.

Sometimes we rather specify the expression. For example, we evaluate the formula for the area of a square with side length \(a\) at \(a = 2\) as follows:

\[A = a^2 = 2^2 = 4.\]

However, we should do it with caution. First, when the substituted value is negative, the number may need to be inside parentheses ( ). For example, \(a+2\) evaluated at \(a=-2\) is simply \(-2+2 = 0.\) However, \(2+a\) evaluated at \(a=-2\) might be \(2+-2,\) which does not make sense. Instead, we have \(2+(-2)=2-2=0.\) Secondly, we must consider implicit multiplications. The implicit multiplication is a simple notation omitting multiplication sign \((\times):\) \(2 \times a = 2a.\) When we evaluate \(2a\) at \(a = 3,\) we must restore the multiplication sign so as to have \(2a = 2 \times 3 = 6,\) not \(2a = 23.\)

## Evaluate \( x^2+13 \) when \( x = 3 \).

Putting \(x=3\) in the given equation, we have:

\[ \begin{align} x^2 + 13 &= 3^2 + 13 \\ &= 9+13 \\ &=22 . \end{align}\]

Thus, the answer is 22. \( _\square \)

## Evaluate \(58+a\) when \( a = -83 \).

Substituting \(a\) with \((-83)\) in the given equation, we have:

\[ \begin{align} 58 + a =& 58 + (-83)\\ =& 58 - 83 \\ =& -25. \end{align}\]

Thus, the answer is -25. \( _\square \)

## Evaluate \(7k \) when \( k = 3 \).

Restoring the multiplication sign, and substituting \(k\) with \(3\) in the given equation, we have:

\[ \begin{align} 7k =& 7 \times k\\ =& 7 \times 3 \\ =& 21. \end{align}\]

Thus the answer is 21. \( _\square \)

## Evaluate \(5m+4n \) when \( m = 3 \) and \(n = -2.\)

We have

\[ \begin{align} 5m + 4n =& 5 \times m + 4 \times n \\ =& 5 \times 3 + 4 \times (-2) \\ =& 15 + (-8) \\ =& 15 - 8 \\ =& 7. \end{align}\]

Thus, the answer is 7. \( _\square \)

## Evaluate \(\frac{x^2}{4}+\frac{y^2}{9} \) when \( x = 4 \) and \(y = -3.\)

We have

\[ \begin{align} \frac{x^2}{4}+\frac{y^2}{9} =& \frac{4^2}{4} + \frac{(-3)^2}{9} \\ =& \frac{16}{4} + \frac{9}{9} \\ =& 4 +1 \\ =& 5. \end{align}\]

Thus, the answer is 5. \( _\square \)

## Evaluate \(\displaystyle{\sqrt{p(q-1)(r-1)}} \) when \( p = 4, \) \(q = -3,\) and \(r=-7.\)

We have

\[ \begin{align} \sqrt{p(q-1)(r-1)} =& \sqrt{4 \times (-3-1) \times (-7-1)} \\ =& \sqrt{4 \times (-4) \times (-8)} \\ =& \sqrt{4 \times 32} \\ =& \sqrt{128} \\ =& 8\sqrt{2}. \end{align}\]

Thus, the answer is \(8\sqrt{2}. \ _\square \)

## Evaluate \(\sqrt{u}\sqrt{v} \) when \( u = -4 \) and \(v=-9.\)

We have

\[ \begin{align} \sqrt{u}\sqrt{v} =& \sqrt{-4}\sqrt{-9} \\ =& -\sqrt{4 \times 9} \\ =& -\sqrt{36} \\ =& -6 . \end{align}\]

Thus, the answer is -6. \(_\square\)

## Evaluate \( 10x + 9x + 8x ..... + x\) , where \( x = 4 \)

We have

\[ \begin{align} 10x + 9x + 8x ..... + x =& x ( 10 + 9 + 8 .... + 1 ) \\ =& x ( 55 ) \\ =& (4) 55 \qquad (\text{since }x=4)\\ =& 220 . \end{align}\]

So, the answer is \(220. \ _\square\)

**Cite as:**Evaluating Expressions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/evaluate-equations-at-specific-values/