Informative line

Operations With Exponents

Evaluation of Power

  • 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:

\(10^3=10×10×10\)

\(10^3=1000\)

Illustration Questions

Which one of the following options is the value of \(4^5\)?

A \(4\)

B \(5\)

C \(20\)

D \(1024\)

×

\(4^5\) can be written as

\(4^5=4×4×4×4×4\)

\(=1024\)

Hence, option (D) is correct.

Which one of the following options is the value of \(4^5\)?

A

\(4\)

.

B

\(5\)

C

\(20\)

D

\(1024\)

Option D is Correct

Addition

  • 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, \(6^2+4^3=100\)

Illustration Questions

Evaluate \(3^2+16\)

A \(25\)

B \(9\)

C \(16\)

D \(40\)

×

\(3^2+16\)

First, evaluate the exponent

\(=9+16\)

Now, add

\(=25\)

Hence, option (A) is correct.

Evaluate \(3^2+16\)

A

\(25\)

.

B

\(9\)

C

\(16\)

D

\(40\)

Option A is Correct

Multiplication

  • 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, \(2^3.3^2=72\)

Illustration Questions

Evaluate: \(6^2.4^3\)

A \(2300\)

B \(64\)

C \(2304\)

D \(36\)

×

\(6^2.4^3\)

First, evaluate the exponents

\(=6×6×4×4×4\)

\(=36×64\)

Now, perform the multiplication

\(=2304\)

Hence, option (C) is correct.

Evaluate: \(6^2.4^3\)

A

\(2300\)

.

B

\(64\)

C

\(2304\)

D

\(36\)

Option C is Correct

Division

  • 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, \(\dfrac{4^2}{2^4}=1\)

Illustration Questions

Evaluate: \(\dfrac{6^3}{3^2}\)

A \(24\)

B \(216\)

C \(9\)

D \(200\)

×

Given: \(\dfrac{6^3}{3^2}\)

First, simplify the exponents.

Multiples of 6 = 1, 2, 3 and 6

So we can write, 

\(\dfrac{6^3}{3^2}=\dfrac{(2×3)^3}{3^2}\)

 

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

\(\dfrac{(2×3)^3}{3^2}=\dfrac{2^3×3^3}{3^2}\)

Now, \(\dfrac{2^3×3^3}{3^2 }=2^3.3^{3-2}\)     \(\Bigg[\because \dfrac{x^a}{x^b}=x^{a-b}\Bigg]\)

\(=2^3.3\)          

Now evaluate, 

\(2^3.3=2×2×2×3\)

\(=24\)

Hence, option (A) is correct.

Evaluate: \(\dfrac{6^3}{3^2}\)

A

\(24\)

.

B

\(216\)

C

\(9\)

D

\(200\)

Option A is Correct

Subtraction

  • 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, \(8^3-36=476\)

Illustration Questions

Evaluate \(9^3-6^3\)

A \(512\)

B \(513\)

C \(216\)

D \(729\)

×

\(9^3-6^3\)

First, evaluate the exponents

\(=9×9×9-6×6×6\)

\(=729-216\)

Now, subtract

\(=513\)

Hence, option (B) is correct.

Evaluate \(9^3-6^3\)

A

\(512\)

.

B

\(513\)

C

\(216\)

D

\(729\)

Option B is Correct

Comparison of Positive Powers with Positive Base

  • 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.

Illustration Questions

Compare: \(6^3\text{_____}5^4\)

A \(<\)

B \(>\)

C \(=\)

D None of these

×

\(6^3\text{_____}5^4\)

Since both base and power are different, so first evaluate the exponent.

\(6^3=6×6×6=216\)

\(5^4=5×5×5×5=625\)

Now, rewrite the values and compare them

\(216<625\)

Hence, option (A) is correct.

Compare: \(6^3\text{_____}5^4\)

A

\(<\)

.

B

\(>\)

C

\(=\)

D

None of these

Option A is Correct

Illustration Questions

What is the greatest whole number that is less than \(\left(\dfrac{3}{4}\right)^3\div\left(\dfrac{3}{6}\right)^2\)?

A \(35\)

B \(5\)

C \(3\)

D \(1\)

×

We have, \(\left(\dfrac{3}{4}\right)^3\div\left(\dfrac{3}{6}\right)^2\)

Using property : \(\left(\dfrac{a}{b}\right)^x=\dfrac{a^x}{b^x}\)

\(=\dfrac{3^3}{4^3}\div\dfrac{3^2}{6^2}\)

\(=\dfrac{3^3}{4^3}×\dfrac{6^2}{3^2}\)     \(\left[\text{By division property},\;\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}×\dfrac{d}{c}\right]\)

\(=\dfrac{3^3×(2×3)^2}{(2×2)^3×3^2}\)  

\(=\dfrac{3^3×2^2×\not3^2}{2^3×2^3×\not3^2}\)        \(\Bigg[\because(x.y)^a=x^a.y^a\Bigg]\)

\(=\dfrac{3^3×2^2}{2^3×2^3}\)

Now evaluate,

\(=\dfrac{3×3×3×\not2×\not2}{2×2×2×2×\not2×\not2}\)

\(=\dfrac{3×3×3}{2×2×2×2}\)

\(=\dfrac{27}{16}=1.6875\)

Thus, the greatest whole number that is less than \(\left(\dfrac{3}{4}\right)^3\div\left(\dfrac{3}{6}\right)^2\) is  \(1\).

Hence, option (D) is correct.

What is the greatest whole number that is less than \(\left(\dfrac{3}{4}\right)^3\div\left(\dfrac{3}{6}\right)^2\)?

A

\(35\)

.

B

\(5\)

C

\(3\)

D

\(1\)

Option D is Correct

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