**Exponents radicals and cube roots :**

Exponent says that how many times do we have to multiply the base by itself.

For example, let us consider

2³ = 2 x 2 x 2 (we multiply the base "2" three times)

Before going to see example based on the above concept, we have to know about two terms

(i) Base

(ii) Exponent (or) power (or) index

**Radicals :**

The square root of a number is the value such that, when a number multiplied by itself, for example

3 x 3 = 9

It is written with a radical sign " √ " and the number or expression inside the radical symbol is called the radicand.

**Cube roots :**

Cube root is inverse operation in finding cubes.

To obtain cube-root of a number, we can use the prime factorization method.

**Step 1 :**

Resolve the given number into prime factors.

**Step 2 :**

Write these factors in triplets such that all three factors in each triplet are equal.

**Step 3 :**

From the product of all factors, take one from each triplet that gives the cube root of a number.

Let us see some examples based on the above concepts.

**Example 1 :**

Simplify 2m^{2} ⋅ 2m^{3}

**Solution :**

= 2m^{2} ⋅ 2m^{3}

= 4m^{(2+3)}

= 4m^{5}

Since we have same base for both terms, we put only one base and add the powers.

Hence the answer is 4m^{5}

**Example 2 :**

Simplify m^{4} ⋅ 2m^{-3}

**Solution :**

= m^{4} ⋅ 2m^{-3}

= 2m^{(4 - 3)}

= 2m^{1}

Since we have same base for both terms, we put only one base and combining the powers.

Hence the answer is 2m.

**Example 3 :**

Simplify (4a^{3})^{2}

**Solution :**

= (4a^{3})^{2}

Since we have power 2 for both the terms 4 and a^{3}, we need to distribute the power for both terms

= 4^{2}(a^{3})^{2}

= 16a^{6}

Hence the answer is 16a^{6}

**Example 4 :**

Simplify (x^{3})^{0}

**Solution :**

= (x^{3})^{0}

Since we have power zero, the answer must be 1.

**Example 5 :**

Simplify 4 a^{3 }b^{2 }⋅ 3 a^{4 }b^{3}

**Solution :**

= 4 a^{3 }b^{2 }⋅ 3 a^{4 }b^{3}

= 12 a^{(3+4)}b^{(2+3)}

= 12 a^{7}b^{5}

Hence the answer is 12 a^{7}b^{5}

**Example 6 :**

Simplify the following

4√3, 18√2, -3√3, 15√2

**Solution :**

** = 4√3 + 18√2 - 3√3 + 15√2**** **

**To simplify the above terms, we need to combine the like terms**

** = 4√3 - 3√3 ****+ 18√2 ****+ 15√2**

** = (4 - 3) √3 ****+ (18 + 15) √2**

** = 1√3 ****+ 33√2**

** = √3 ****+ 33√2**

**Let us look into the next example on "**Exponents radicals and cube roots"

**Example 7 :**

Simplify the following

2∛2, 24∛2, - 4∛2

**Solution :**

** = **2∛2 + 24∛2 - 4∛2

** = (**2 + 24 - 4) ∛2

** = **22 ∛2

Let us look into the next example on "Exponents radicals and cube roots"

**Example 8 :**

Multiply ∛13 x ∛5

**Solution :**

** = ∛13 x ∛5 **

**Since the index of both root terms are same, we can write only one root sign and multiply the numbers.**

** = ∛(13 x 5) **

** = ∛65**

**Example 9 :**

Multiply 15√54 ÷ 3√6

**Solution :**

** = **15√54 ÷ 3√6

**Since the index of both root terms are same, we can write only one root and divide the numbers.**

**= (**15/3)√(54/6)

**= **5√9 ==> 5√(3 x 3) ==> 5 x 3 ==> 15

**Example 10 :**

Multiply (48)^{1/4 }÷ (72)^{1/8}

**Solution :**

** = **(48)^{1/4 }÷ (72)^{1/8}

**Since the index of the above root terms are not same, we need to convert the power 1/4 as 1/8.**

** = **(48)^{(}^{1/4) x (2/2) }÷ (72)^{1/8}

** = **(48)^{(2}^{/8) }÷ (72)^{1/8}

** = **48^{2 (1}^{/8) }÷ (72)^{1/8}

**= [(**48 x 48)^{ }÷ (72)]^{1/8}

**= [2304**^{ }÷ 72]^{1/8}

**= (32)**^{1/8}

**Let us look into the next example on "**Exponents radicals and cube roots"

**Example 11 :**

Find the cube-root of 512

**Solution :**

Hence cube-root of 512 is 8.

**Example 12 :**

Find the cube-root of 27 x 64

**Solution :**

= ∛27 x 64

We can write 27 as 3 x 3 x 3, like wise 64 as 4 x 4 x 4.

= ∛3 x 3 x 3 x 4 x 4 x 4

= 3 x 4

= 12

Hence the answer is 12.

**Example 13 :**

Find the cube-root of 125/216

**Solution :**

Here we need to find the cube-root for a fraction. For that, split the numerator and denominator as much as possible.

= ∛125/216

125 = 5 x 5 x 5 and 64 = 4 x 4 x 4

= ∛(5 x 5 x 5) /(4 x 4 x 4)

Since we have cube-root, we need to take one for each three same terms.

= 5/4

Hence the cube root of 125/216 is 5/4.

**Example 14 :**

Find the cube-root of -512/1000

**Solution :**

Here we need to find the cube-root for a fraction. In the cube-root we have negative sign.

Whenever we have negative sign inside the cube-root, the answer must have negative sign.

= ∛512/1000

512 = 8 x 8 x 8 and 1000 = 10 x 10 x 10

= - ∛(8 x 8 x 8)/(10 x 10 x 10)

Since we have cube-root, we need to take one for each three same terms.

= - 8/10

If it is possible, we may simplify

= - 4/5

Hence the cube-root of ∛-512/1000 is -4/5.

**Example 15 :**

Find the cube-root of 0.027

**Solution :**

Here we need to find the cube-root for a decimal.

First let us convert the given decimal as fraction. For that, we have to multiply and divide by 1000.

0.027 x (1000/1000) = 27/1000

∛0.027 = ∛27/1000

= ∛(3 x 3 x 3)/(10 x 10 x 10)

= 3/10

Hence the cube-root of ∛0.027 is 3/10.

- Properties of exponents
- Evaluating exponents
- Evaluate integers raised to the rational exponents
- Properties of square roots
- Square roots
- Ordering square roots from least to greatest
- Operations with roots
- Simplifying expressions involving square roots worksheets
- Evaluate radical expression
- Solve equations involving square roots
- Simplifying expressions with conjugates
- Simplify expressions with variables
- Solving equations with square root signs on both sides
- Estimating square roots
- Finding square roots and cube roots of a number

After having gone through the stuff given above, we hope that the students would have understood "Exponents radicals and cube roots".

If you want to know more about the stuff "Exponents radicals and cube roots", please click here

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