Problem 568 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(5^{\frac{1}{2}} \cdot 5^{\frac{7}{2}}\) (b) \(c^{\frac{3}{4}} \cdot c^{\frac{9}{4}}\) c) \(d^{\frac{3}{5}} \cdot d^{\frac{2}{5}}\)

Step-by-Step Solution

Answer

(a) \(5^4\), (b) \(c^3\), (c) \(d\)

Step 1 - Use properties of exponents for part (a)

Combine the exponents when multiplying like bases. For part (a), this means applying the property: \(a^m \times a^n = a^{m+n}\). Here, \(m = \frac{1}{2}\) and \(n = \frac{7}{2}\). So, \(5^{\frac{1}{2}} \cdot 5^{\frac{7}{2}} = 5^{\frac{1}{2} + \frac{7}{2}}\).

Step 2 - Simplify the exponent for part (a)

Add the exponents together: \[5^{\frac{1}{2} + \frac{7}{2}} = 5^{\frac{1+7}{2}} = 5^{\frac{8}{2}} = 5^4\].

Step 3 - Use properties of exponents for part (b)

Like in part (a), combine the exponents: \( c^{\frac{3}{4}} \cdot c^{\frac{9}{4}} = c^{\frac{3}{4} + \frac{9}{4}} \).

Step 4 - Simplify the exponent for part (b)

Add the exponents together: \[ c^{\frac{3}{4} + \frac{9}{4}} = c^{\frac{3+9}{4}} = c^{\frac{12}{4}} = c^3 \].

Step 5 - Use properties of exponents for part (c)

Similarly, combine the exponents: \( d^{\frac{3}{5}} \cdot d^{\frac{2}{5}} = d^{\frac{3}{5} + \frac{2}{5}} \).

Step 6 - Simplify the exponent for part (c)

Add the exponents together: \[ d^{\frac{3}{5} + \frac{2}{5}} = d^{\frac{3+2}{5}} = d^{\frac{5}{5}} = d^1 = d \].

Key Concepts

properties of exponents

When working with exponents, understanding their properties can simplify many calculations. One key property is the multiplication rule: \(a^m \times a^n = a^{m+n}\).
This means when you multiply two expressions with the same base, you add their exponents together.
For example, in part (a) of the exercise, we used this property to combine \(5^{1/2} \times 5^{7/2}\).
By adding the exponents, we get \(5^{(1/2 + 7/2)} = 5^4\).
It's important to simplify fractions when dealing with exponents to easily manage the calculations.

combining exponents

Combining exponents correctly is crucial when simplifying expressions.
Once you know the properties, adding and rearranging exponents becomes straightforward.
In part (b) of the exercise, we dealt with \(c^{3/4} \times c^{9/4}\).
We combined the exponents by adding them: \(\frac{3}{4} + \frac{9}{4} = \frac{12}{4} = 3\).
Thus, \(c^{3/4} \times c^{9/4} = c^3\).
Always ensure to combine fractions properly and simplify them to the lowest terms.

like bases

Understanding 'like bases' is essential in working with exponents. Like bases mean the base numbers in exponential expressions are the same.
For example, in part (c) of the exercise, we simplified expressions with like bases \(d^{3/5} \times d^{2/5}\).
We combined them by adding the exponents, resulting in \(d^{(3/5 + 2/5)} = d^1 = d\).
Recognizing like bases helps in easily applying exponent properties and simplifying expressions efficiently.