Evaluate (7^(3/2))/(7^(1/2))
The question is asking for the simplification of a mathematical expression that involves exponents. Specifically, it involves dividing two powers of the number 7, where the first power is 7 raised to three halves and the second power is 7 raised to one half. The operation needed here is to apply the laws of exponents for division, which state that when you divide like bases, you subtract the exponents. In this case, the bases are the same (both are 7), and you have to subtract the exponent in the denominator (1/2) from the exponent in the numerator (3/2).
$\frac{7^{\frac{3}{2}}}{7^{\frac{1}{2}}}$
Answer
Solution:
Step 1:
Apply the negative exponent rule to rewrite the denominator as a negative exponent in the numerator: $7^{\frac{3}{2}} \cdot 7^{-\frac{1}{2}}$.
Step 2:
Combine the exponents by adding them together.
Step 2.1:
Utilize the property of exponents that states $a^{m} \cdot a^{n} = a^{m+n}$ to merge the exponents: $7^{\frac{3}{2} - \frac{1}{2}}$.
Step 2.2:
Add the exponents by finding a common denominator: $7^{\frac{3 - 1}{2}}$.
Step 2.3:
Perform the subtraction in the numerator: $7^{\frac{2}{2}}$.
Step 2.4:
Simplify the fraction within the exponent by dividing: $7^{1}$.
Step 3:
Conclude by simplifying the expression to $7^{1}$, which equals $7$.
Knowledge Notes:
The problem involves simplifying an expression with exponential notation, specifically with fractional exponents. The key knowledge points and rules used in the solution are:
Negative Exponent Rule: The rule $\frac{1}{b^{n}} = b^{-n}$ allows us to rewrite a fraction with an exponent in the denominator as a multiplication with a negative exponent in the numerator.
Power Rule for Exponents: The rule $a^{m} \cdot a^{n} = a^{m+n}$ states that when multiplying like bases, you add the exponents.
Simplifying Fractions: When you have a fraction as an exponent, you can simplify it just as you would any other fraction.
Simplification of Exponents: An exponent of $1$, such as $a^{1}$, simply equals $a$ since any number to the power of one is itself.
Combining these rules allows for the simplification of expressions involving exponents, particularly when the bases are the same and only the exponents differ.
