What are the properties of rational exponents and how are they used tk solve problems
The Correct Answer and Explanation is:
Rational exponents are exponents that are expressed as fractions. They are a way to extend the idea of exponents to include roots, and they follow a set of properties that allow them to be manipulated in similar ways as integer exponents. Rational exponents are often used in algebra and other areas of mathematics to simplify expressions and solve equations involving roots.
Properties of Rational Exponents:
- General Form:
A rational exponent is written as ( a^{\frac{m}{n}} ), where ( m ) is the numerator and ( n ) is the denominator. This expression can be interpreted as:
[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
]
This means that the base ( a ) is raised to the power of ( m ) and then the ( n )-th root is taken. Alternatively, it can be written as ( ( \sqrt[n]{a} )^m ), where the base ( a ) is first rooted and then raised to the power ( m ). - Product Rule:
For any positive real numbers ( a ) and ( b ) and rational exponents ( \frac{m}{n} ) and ( \frac{p}{q} ), the product rule is:
[
a^{\frac{m}{n}} \times b^{\frac{p}{q}} = (a \times b)^{\frac{m}{n} + \frac{p}{q}}
]
This allows you to add the exponents when multiplying like terms with rational exponents. - Quotient Rule:
For division, the quotient rule is:
[
\frac{a^{\frac{m}{n}}}{b^{\frac{p}{q}}} = \frac{a}{b}^{\frac{m}{n} – \frac{p}{q}}
]
This allows you to subtract exponents when dividing terms with rational exponents. - Power Rule:
When raising a number with a rational exponent to another power, use the rule:
[
\left( a^{\frac{m}{n}} \right)^r = a^{\frac{m \times r}{n}}
]
Solving Problems with Rational Exponents:
Rational exponents simplify the process of solving equations involving roots. For example, to solve ( x^{\frac{2}{3}} = 16 ), we can eliminate the fractional exponent by raising both sides to the reciprocal of ( \frac{2}{3} ), which is ( \frac{3}{2} ). Thus:
[
\left( x^{\frac{2}{3}} \right)^{\frac{3}{2}} = 16^{\frac{3}{2}}
]
[
x = 16^{\frac{3}{2}} = \sqrt{16^3} = \sqrt{4096} = 64
]
So, ( x = 64 ).
In summary, rational exponents are a powerful tool for working with powers and roots in algebra. Their properties—such as the product, quotient, and power rules—make them useful for simplifying complex expressions and solving equations involving powers and roots.