Informative line

# 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:

$$10^3=10×10×10$$

$$10^3=1000$$

#### 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

• 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

$$=36+64$$

$$=100$$

Thus, $$6^2+4^3=100$$

#### Evaluate $$3^2+16$$

A $$25$$

B $$9$$

C $$16$$

D $$40$$

×

$$3^2+16$$

First, evaluate the exponent

$$=9+16$$

$$=25$$

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, $$2^3.3^2=72$$

#### 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, $$\dfrac{4^2}{2^4}=1$$

#### 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, $$8^3-36=476$$

#### 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.

#### 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

#### 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