• Evaluation of power means solving it.
• We can find the value of a power by completing the multiplication and finding the new product.

For example: (i) \(7^3\)

By evaluating, we get:

\(7^3=7×7×7\)

\(=49×7\)

\(=343\)

(ii) \(10^3\)

By evaluating, we get:

\(10^3=10×10×10\)

\(10^3=1000\)

• We can apply different operations on powers.
• We can add the terms having powers.
• For this, first, evaluate each exponent and then perform the operation.

For example: \(6^2+4^3\)

First, we will evaluate the exponents

\(=36+64\)

Now, we will add

\(=100\)

Thus, \(6^2+4^3=100\)

• We can apply different operations on powers.
• We can multiply the different terms having different powers.
• For this, first, evaluate the exponents and then multiply them.

For example:

\(2^3.3^2\)

First, we will evaluate the exponents.

\(=2×2×2×3×3\)

\(=8×9\)

Now, we will multiply

\(=72\)

Thus, \(2^3.3^2=72\)

• We can apply different operations on powers.
• We can divide different terms having different powers.
• For this, first, evaluate the exponents and then divide them.

For example: \(\dfrac{4^2}{2^4}\)

First, simplify the exponents.

Multiples of 4 = 1, 2 and 4 

So we can write,

 \(\dfrac{4^2}{2^4}=\dfrac{(2×2)^2}{{2}^4}\)

Using the property \((x.y)^a=x^a.y^a\)

\(=\dfrac{2^2×2^2}{2^4}\)

\(=\dfrac{{2}^{(2+2)}}{{2}^4}\)     \(\Bigg[\because\, x^a.x^b=x^{a+b}\Bigg]\)

\(=\dfrac{2^4}{2^4}\)      

Now, divide 

\(=1\)

Thus, \(\dfrac{4^2}{2^4}=1\)

• We can apply different operations on powers.
• We can subtract the terms having powers.
• For this, first, evaluate each exponent and then perform the operation.

For example: \(8^3-36\)

First, we will evaluate the exponent

\(=512-36\)

Now, we will subtract

\(=476\)

Thus, \(8^3-36=476\)

• We can show the comparison between two numbers having powers by using greater than, less than or equal to signs.
• For comparison, we use the following three symbols:

       > Greater than

      < Less than

      = Equal to

• There can be three cases while comparing the numbers having positive powers.

Case 1: When the base is same but the power is different.

For example: \((4)^2\text{_____}(4)^3\)

Since the base is same, so we will compare the powers.

We know that \(2<3\)

Thus,

\((4)^2<(4)^3\)

Hence, when the base is same but the power is different, then the number with larger power is greater.

Case 2: When the power is same but the base is different.

For example: \((5)^3\text{_____}(10)^3\)

Since the power is same, so we will compare the bases.

We know that \(5<10\)

Thus, \((5)^3<(10)^3\)

Hence, when the power is same but the base is different, then the number with larger base is greater.

Case 3: When both base and power are different.

For example: \((5)^3\text{_____}(6)^2\)

First, we will evaluate the terms.

\(5^3=5×5×5=125\)

\(6^2=6×6=36\)

Now, we will rewrite the numbers and compare them.

\(125>36\)

Hence, when both power and base are different, we first evaluate each exponent and then compare them.

Note: The opening of the symbol is always towards the larger number.