Simplifying Exponents

Simplifying exponents means reducing the algebraic expressions with exponents to their simplest form. We use certain rules to simplify those expressions.

With the Same Base

When simplifying exponents with the same base, we follow the rules:

• x^m × xⁿ = x^{m + n} 
• x^m ÷ xⁿ = x^{m – n} 

Let us simplify y⁷ × y³ 

= (y × y × y × y × y × y × y) × (y × y × y), which leaves a total of 10 y’s.

Thus, the simplified expression becomes y¹⁰ 

However, using the product law, it can be easily simplified 

y⁷ × y³ = y^{7 + 3} = y¹⁰ 

With Different Bases

There can be two possible cases when multiplying or dividing exponents with different bases. 

Different Bases and the Same Power

When simplifying exponents with different bases and the same power, we follow the given rules:

Here, we follow the rules:

• x^m × y^m = (xy)^m 
• x^m ÷ y^m = (x ÷ y)^m 

Let us simplify p⁴ × q⁴ 

= (p × p × p × p) × (q × q × q × q)

= (p × q) × (p × q) × (p × q) × (p × q), which leaves a total of 4 (p × q)’s.

Thus, the simplified expression becomes (p × q)⁴ 

However, using the power of product law, it is easily simplified as: 

p⁴ × q⁴ = (p × q)⁴ 

Different Bases and Different Powers

When simplifying exponents with different bases and different powers, we follow the given rules:

• x^m × yⁿ = x^myⁿ 
• x^m ÷ yⁿ = ${\dfrac{x^{m}}{y^{n}}}$

Let us simplify 3⁴ ÷ 7² 

= (3 × 3 × 3 × 3) ÷ (7 × 7)

= ${\dfrac{81}{49}}$

Thus, the simplified expression becomes ${\dfrac{81}{49}}$ 

Simplifying Fractional Exponents

When simplifying fractional exponents, we follow the following rules:

• ${x^{\dfrac{m}{n}}\times x^{\dfrac{p}{q}}=x^{\left( \dfrac{m}{n}+\dfrac{p}{q}\right) }}$
• ${x^{\dfrac{m}{n}}\times y^{\dfrac{m}{n}}=\left( x\times y\right) ^{\dfrac{m}{n}}}$
• ${x^{\dfrac{m}{n}}\times y^{\dfrac{m}{n}}=x^{\dfrac{m}{n}}y^{\dfrac{m}{n}}}$
• ${x^{\dfrac{m}{n}}\div x^{\dfrac{p}{q}}=x^{\left( \dfrac{m}{n}-\dfrac{p}{q}\right) }}$
• ${x^{\dfrac{m}{n}}\div y^{\dfrac{m}{n}}=\left( \dfrac{x}{y}\right) ^{\dfrac{m}{n}}}$
• ${x^{\dfrac{m}{n}}\div y^{\dfrac{p}{q}}=\dfrac{x^{\dfrac{m}{n}}}{y^{\dfrac{p}{q}}}}$

Let us simplify ${8^{\dfrac{2}{3}}\div 4^{\dfrac{3}{2}}}$

= ${\dfrac{8^{\dfrac{2}{3}}}{4^{\dfrac{3}{2}}}}$

= ${\dfrac{\left( \sqrt[3] {8}\right) ^{2}}{\left( \sqrt{4}\right) ^{3}}}$

= ${\dfrac{\left( 2\right) ^{2}}{\left( 2\right) ^{3}}}$

= ${2^{2-3}}$

= ${2^{-1}}$

= ${\dfrac{1}{2}}$

Thus, the simplified expression is ${\dfrac{1}{2}}$

Simplifying Negative Exponents

When simplifying fractional exponents, we follow the following rules:

• ${x^{-m}\times x^{-n}=x^{-\left( m+n\right) }=\dfrac{1}{x^{m+n}}}$
• ${x^{-m}\times y^{-m}=\left( x\times y\right) ^{-m}=\dfrac{1}{\left( x\times y\right) ^{m}}}$
• ${x^{-m}\times y^{-n}=x^{-m}y^{-n}=\dfrac{1}{x^{-m}y^{-n}}}$
• ${x^{-m}\div x^{-n}=x^{-\left( m-n\right) }=\dfrac{1}{x^{m-n}}}$
• ${x^{-m}\div y^{-m}=\left( \dfrac{x}{y}\right) ^{-m}=\left( \dfrac{y}{x}\right) ^{m}}$
• ${x^{-m}\div y^{-n}=\dfrac{x^{-m}}{y^{-n}}=\dfrac{y^{n}}{x^{m}}}$

Let us simplify 4³ × 4^-1 

= 4^{3 + (-1)} 

= 4² 

Thus, the simplified expression is 4² 

Solved Examples