Basically, the exponents state the repeated multiplication of the same number with respect to the number of times. For example, 23 = 2 x 2 x 2.Therefore, using exponents means raising a number to a power, where the exponent is the power. The number to which the power is raised is called the base of the power, thus, in 23, 2 is the base and 3 is the exponent. The difference between the power and exponents are, the exponent is the power raised to the base number and power is represented as the combination of base number and exponent. Thus, 23 represents power.

To denote the exponent value, the base number value should be the same and then altogether it is said to be a power. Exponents and Powers are used to express large numbers which cannot be represented in the general form, such as 100000000000000 can be represented as 1014.

## Exponents and Powers Formulas

Let us discuss here the basic formulas of exponents and powers for Class 7 standard.

If p is a rational number and have a non-zero value, m is a natural number, then,

p × p × p × p ×…..× p(m times) is written as pm, where p is the base number and m is the exponent value and pm is the power and ‘pm’ is said as ‘p raised to the power m’. This is the general representation of exponents and powers.

Example: 9 × 9 × 9 × 9 × 9 × 9 × 9 = 97 , , where 9 is the base number and 7 is the exponent.

There are key to laws of exponents defined to solve complex problems based on powers and exponents.

### Laws of Exponents

If p and q are non- zero rational numbers and m and n are natural numbers, then the laws of exponents can be explained as;

• pm x pn = pm+n
• (pm)n = pmn
• $\frac{p^m}{p^n}$ = pm-n ; where m>n
• $\frac{p^m}{p^n}$ = 1/pn-m ; where n>m
• (pq)m = pm qm
• $(\frac{p}{q})^m$ = $\frac{p^m}{q^m}$

### Exponent and Powers Example Problems

Problem: Solve 23 x 22

Solution: From the law of exponent we know,

pm x pn = pm+n

Therefore, 23 x 22 = 23+2 = 25 = 2 x 2 x 2 x 2 x 2 = 32

Problem: Solve $\frac{3^3}{3^2}$

Solution: Applying the law of exponent, we get,

$\frac{p^m}{p^n}$ = pm-n

So, $\frac{3^3}{3^2}$ = 33-2 = 31 = 3

Problem: Solve (52)2

Solution: We know, by the law of exponent,

(pm)n = pmn

Therefore, (52)2 = 52×2 = 54 = 625

Problem: What is the answer for $(\frac{10}{5})^2$ ?

Solution: By the law of exponent, we can write the given equation as;

$(\frac{10}{5})^2$ = $\frac{10^2}{5^2}$

= $\frac{100}{25}$

= 4

Problem: If $\frac{2^3}{2^5}$ is the given equation. Find the answer.

Solution: From the given problem, we can see the power of the denominator is greater than the power in the numerator.

Therefore, by the law of exponent,

$\frac{p^m}{p^n}$ = 1/pn-m ; where n>m $\frac{2^3}{2^5}$

= 1/25-3

= 1/22

= ¼