Powers and Roots

$5^3=5\times5\times5=125,$

$5$ is the base and $3$ is the index (also called the exponent or power). The index tells you how many factors of the base are multiplied together. It does not mean “multiply the base by the index”: $5^3$ is not $5\times3$.

A positive integer index is a whole-number index greater than zero. For example,

$7^1=7,\qquad 7^2=49,\qquad 7^3=343.$

The names square and cube come from geometry. A square of side $a$ has area $a^2$, while a cube of edge $a$ has volume $a^3$. Powers such as $a^4$ and $a^5$ are read as “$a$ to the power of four” and “$a$ to the power of five”.

Brackets matter when the base is negative:

$(-2)^4=16\qquad\text{but}\qquad (-2)^5=-32.$

An even number of negative factors gives a positive result; an odd number gives a negative result.

Recognising powers of 2, 3, 4 and 5

Knowing common powers makes calculations and estimates much quicker. Each entry is found by multiplying the previous one by the base.

Some numbers appear in more than one row. For example,

$64=2^6=4^3.$

This happens because $4=2^2$, so every power of $4$ is also an even-numbered power of $2$: $4^n=2^{2n}$. Recognising such relationships helps when working backwards from a number to its base and index.

Roots reverse powers

A root asks the inverse question: “Which number was raised to this power?” Since

$4^3=64,$

we have

$\sqrt[3]{64}=4.$

The small number on the radical is the type of root. Thus $\sqrt[3]{a}$ is a cube root, $\sqrt[4]{a}$ is a fourth root and $\sqrt[n]{a}$ is an $n$th root. For square roots, the index $2$ is normally omitted:

$\sqrt{81}=9.$

For a positive number $a$, the radical $\sqrt[n]{a}$ denotes its positive, or principal, $n$th root. This must be distinguished from solving an equation. Although $\sqrt{81}=9$, the equation $x^2=81$ has two real solutions, $x=9$ and $x=-9$, because both square to $81$.

The same distinction applies to any even power. For example, $\sqrt[4]{16}=2$, but $x^4=16$ has the real solutions $x=\pm2$. An odd power keeps the sign of its base, so an odd root of a negative number can be real: $\sqrt[5]{-32}=-2$. By contrast, an even root of a negative number is not a real number.

Estimating roots by bracketing

When a number is not a perfect power, place it between two nearby perfect powers. To estimate $\sqrt{70}$, use

$8^2=64<70<81=9^2.$

Therefore,

$8<\sqrt{70}<9.$

Because $70$ is between $64$ and $81$, a sensible one-decimal estimate is $8.4$. Checking gives $8.4^2=70.56$, which is close to $70$. The check also shows that the exact root is slightly less than $8.4$.

The same method works for higher roots. Since

$3^3=27<50<64=4^3,$

it follows that $3<\sqrt[3]{50}<4$. Trying $3.7$ gives $3.7^3=50.653$, so $\sqrt[3]{50}\approx3.7$ to one decimal place.

The gaps between powers are not evenly spaced, so simply choosing the midpoint of the two bases need not give the best estimate. Test a sensible decimal and raise it to the required power; then adjust it up or down.

Estimating powers

To estimate a power of a positive number, replace the base with a nearby convenient value or first establish bounds. For example, $3<3.7<4$, so

$3^3<3.7^3<4^3,$

and hence

$27<3.7^3<64.$

As a rough estimate, $3.7^3\approx50$ is reasonable. Indeed, multiplying $3.7\times3.7\times3.7$ gives about $50.7$, confirming the estimate.

The required accuracy determines the method. Rounding $3.7$ to $4$ gives the quicker estimate $4^3=64$, but it is much rougher. When an answer must be more precise, keep more digits in the base or test nearby decimal values. Whatever method you use, compare the estimate with known powers so that its size is plausible.