**Simplify expressions involving rational exponents :**

Here we are going to see some practice questions on simplifying expressions involving rational exponents.

**Rule 1 :**

If we have same base for two or more terms which are multiplying , we have to write only one base and add the powers

**a ^{m} x a^{n} = a^{(m+n)}**

**Rule 2 :**

If we have same base for two or more terms which are dividing, we have to write only one base and subtract the powers

**a ^{m} / a^{n} = a^{(m-n)}**

**Rule 3 :**

If we want to change the negative power as positive, we have to take its reciprocal of the base and change the sign.

**a ^{-m} = 1/a^{m}**

**(a/b) ^{-m} = (b/a)^{m}**

**Let us look into some example problems based on the above concept.**

**Example 1 :**

Simplify the following expression

**Solution :**

= (3x^{-1/2} ⋅ 3x^{1/2 }^{⋅ }y^{-1/3) }/ 3y^{-7/4}

= 9x^{(}^{-1/2) + (}^{1/2) }⋅ y^{(}^{-1/3) + (}^{7/4)}/ 3

= (9x^{0}^{ }⋅ y^{(}^{-4+21)/12})/3

= (9(1) ⋅ y^{17}^{/12})/3

= 3y^{17}^{/12}

**Step 1 :**

We have to combine the x and y terms separately.

3x^{-1/2} ⋅ 3x^{1/2 }= 9x^{(}^{-1/2) + (}^{1/2) }

Since we bring the y term from denominator to numerator, we have changed the original sign from negative to positive.

y^{-1/3 }/ y^{-7/4 }= y^{(}^{-1/3) + (}^{7/4)}

**Step 2 :**

By combining the x terms, we get x^{0}

For combining the y terms, we need to take L.C.M. So we get y^{17}^{/12}

**Step 3 :**

Hence the answer is 3y^{17}^{/12}

**Example 2 :**

Simplify the following expression

**Solution :**

= 3y^{1/4} /( 4x^{-2/3} ⋅ y^{3/2 }^{⋅ }3y^{1/2) }

= 3y^{1/4} / (12 x^{-2/3} ⋅ y^{3/2 }y^{1/2) }

= y^{1/4} / (4 x^{-2/3} ⋅ y^{(3/2) + (}^{1/2)})

= y^{1/4} / (4 x^{-2/3} ⋅ y^{(3+1)/2})

= y^{1/4} / (4 x^{-2/3} ⋅ y^{2})

= x^{2/3}/ (4 ⋅ y^{(-}^{1/4) +}^{2})

= x^{2/3}/ (4 ⋅ y^{(-}^{1 + 8)/4})

= x^{2/3}/ (4 ⋅ y^{7}^{/4})

**Step 1 :**

In the denominator, we multiply 4 and 3.

So we get 12 x^{-2/3} ⋅ y^{3/2 }y^{1/2}

In the denominator, we get y terms.So we get

y^{(3/2) + (}^{1/2) }= y^{2}

**Step 2 :**

Hence the answer is x^{2/3}/ (4 ⋅ y^{7}^{/4})

**Example 3 :**

Simplify the following expression

**Solution :**

To simplify the above expression, we have to use exponent rules.

**Step 1 :**

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

**Step 2 :**

By considering the above fractions 2/3 and 7/3, we have same denominators for both fractions. So we dont have to take L.C.M

**Step 3 :**

Hence the simplified answer is y³

**Example 4 :**

Simplify and write the answer in positive exponents

**Solution :**

To simplify the above expression, we have to use exponent rules.

**Step 1 :**

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

**Step 2 :**

By considering the above fractions 3/5 and 7/5, we have same denominators for both fractions. So we dont have to take L.C.M

**Step 3 :**

Hence the simplified answer is a²

**Example 5 :**

Simplify and write the answer in positive exponents

**Solution :**

To simplify the above expression, we have to use exponent rules.

**Step 1 :**

Here we have common power for both terms, so we have to distribute the power 1/2.

**Step 2 :**

Whenever we have power raised to another power, we have to multiply both powers.

**Step 3 :**

By simplifying 4 and 2, we get x²

Hence the simplified answer is x² y^(1/2)

**Example 6 :**

Simplify and write the answer in positive exponents

**Solution :**

To simplify the above expression, we have to use exponent rules.

**Step 1 :**

Here we have common power for both terms, so we have to distribute the power 2.

**Step 2 :**

Whenever we have power raised to another power, we have to multiply both powers.

**Step 3 :**

By simplifying 2 and 2, we get a

Hence the simplified answer is a b^(2/3)

**Example 7 :**

Simplify and write the answer in positive exponents

**Solution :**

To simplify the above expression, we have to use exponent rules.

**Step 1 :**

In the first step we have multiplied the numbers.

**Step 2 :**

Now we have to multiply a^1/2 and a. Since both are having same base, we have to put only one base and add the powers.

**Step 3 :**

By adding a^1/2 with a^1, we get a^3/2.

Hence the simplified answer is 6 a^(3/2)

After having gone through the stuff given above, we hope that the students would have understood "Simplify expressions involving rational exponents".

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