Evaluating Expressions: Powers, Roots and Absolute Value

Two habits keep substitution safe.

• Write the value in parentheses. For $x = 5$, write $3(5)$, not $35$, so the multiplication stays visible. For $x = -2$, write $3(-2)$, so the negative sign stays attached to the $2$.
• Keep the order of operations. Multiply and divide from left to right before you add and subtract from left to right. That is why every row of the table multiplies first.

In the middle row, $3(0) = 0$ because zero times any number is zero. The value of the expression is still $-4$, because subtracting $4$ is still part of the rule.

A power is a short way to write repeated multiplication. In $a^n$, the base $a$ is the number you multiply, and the exponent $n$ tells how many copies of the base are multiplied together. Each copy is called a factor. In this lesson the exponent is a positive whole number, so $2^3 = 2 \cdot 2 \cdot 2 = 8$ and $x^2 = x \cdot x$. The second power of a number is its square and the third power is its cube.

The result of a multiplication is a product, and its sign depends on its factors. Negative factors pair up, and each pair multiplies to a positive number. So when no factor is zero, an even number of negative factors gives a positive product, and an odd number gives a negative product. That is why $(-2)^4 = 16$ is positive and $(-2)^3 = -8$ is negative.

An exponent belongs only to the number, variable or parentheses directly in front of it. In $(-3)^2$ the parentheses make $-3$ the base. In $-3^2$ the base is only $3$, and the negative sign takes the opposite of the result afterwards. The table compares the two.

A root asks a power question in reverse: which number, squared or cubed, gives this result?

The square root of $n$, written $\sqrt{n}$, is the number that is zero or positive and whose square is $n$. It is defined for $n \ge 0$. So $\sqrt{25} = 5$, because $5^2 = 25$ and $5$ is positive. It is true that $(-5)^2 = 25$ too, but the symbol $\sqrt{25}$ names only $5$. A negative number has no real square root, because squaring any real number, a number on the number line, gives zero or a positive result.

The cube root of $n$, written $\sqrt[3]{n}$, is the number whose cube is $n$. Every real number has exactly one real cube root: positive for a positive number, negative for a negative number, and $0$ for $0$. So $\sqrt[3]{8} = 2$ because $2^3 = 8$, and $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$.

A root symbol also groups. Work out everything under it before you take the root. Every root in this lesson comes out exactly, so no rounding is needed.

The absolute value of a number is its distance from $0$ on the number line. A distance is never negative, so an absolute value is never negative. You write it with two vertical bars: $\lvert 5 \rvert = 5$ and $\lvert -5 \rvert = 5$, because $5$ and $-5$ are each $5$ units from $0$. Also, $\lvert 0 \rvert = 0$.

In an expression, the bars are grouping symbols. First find the single number between the bars, then take its distance from $0$. Anything outside the bars waits. In $-\lvert x \rvert$ the negative sign is outside, so it takes the opposite after the absolute value is found.

The table evaluates five expressions for $x = -7$. Compare the second and third rows: moving the $3$ outside the bars changes the result.

Takeaway. To evaluate an expression, substitute each given value in parentheses, then compute in order: grouping symbols first, then powers and roots, then multiplication and division, then addition and subtraction. Parentheses protect a negative value, so $(-3)^2 = 9$ while $-3^2 = -9$. For $n \ge 0$, the symbol $\sqrt{n}$ names only the root that is zero or positive. The cube root of a negative number is negative. An absolute value is a distance from $0$, so it is never negative.