Question

Evaluate the expression. Write fractional answers in simplest form.${(-\frac{3}{5})}^3{\left(\frac{5}{3}\right)}^2$

Short answer

-3/5

Step-by-step solution

Evaluate the first part of the expression

First, evaluate ${(-\frac{3}{5})}^3$. When a negative number is raised to an odd power, the result will be negative. When we raise a fraction to a power, both the numerator and the denominator are raised to that power. Thus, \( \left(-\frac{3}{5}\right)^{3} = -\left(\frac{3}{5}\right)^{3} = -\frac{3^{3}}{5^{3}} = -\frac{27}{125}.

Evaluate the second part of the expression

Next, evaluate ${\left(\frac{5}{3}\right)}^2$. When we raise a fraction to a power, both the numerator and the denominator are raised to that power. Plus, when a positive number is raised to an even power, the result will also be positive. Thus, \( \left(\frac{5}{3}\right)^{2} = \frac{5^{2}}{3^{2}} = \frac{25}{9}

Evaluate the entire expression

Finally, multiply the two results together. So, $(-\frac{27}{125})\ast \left(\frac{25}{9}\right)$ equals $-\frac{27}{125}\ast \frac{25}{9}=-\frac{675}{1125}$. However, this is not in the simplest form. The simplest form can be found by dividing both the numerator and the denominator by their greatest common divisor which is 75. So the simplest form is $-\frac{9}{15}$

Simplify the answer

The fraction $-\frac{9}{15}$ can be simplified further by dividing both the numerator and the denominator by their greatest common divisor which is 3. Thus, $-\frac{9}{15}$ simplifies to $-\frac{3}{5}$.

Key concepts

Fractional Exponents

Fractional exponents are a way to express powers and roots in a unified manner. The expression of an exponent as a fraction means that the numerator indicates the power to which the base number is raised, while the denominator represents the type of root.
For example, the expression $a^{\frac{m}{n}}$ implies that the base $a$ is first raised to the power of $m$ and then the nth root is taken. In practice, this is equivalent to taking the nth root of $a$ first and then raising the result to the power of $m$ (or vice versa, the order of operations does not affect the outcome).
Understanding how to work with these exponents is vital, especially when dealing with complex algebraic expressions. Remember to pay attention to the signs of the base and the value of the exponent as they dictate whether the result will be positive or negative, as seen in our exercise example where negative base raised to an odd power resulted in a negative value.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics as it helps to make calculations easier and results more understandable. To simplify a fraction, you find the greatest common divisor (GCD), also known as the greatest common factor (GCF), between the numerator and the denominator and divide both by this number.
Take, for instance, the fraction $\frac{675}{1125}$ we encountered in our exercise. By determining that both numbers are divisible by 75, we simplify the fraction to $\frac{9}{15}$ by dividing both the numerator and the denominator by 75. However, this may not always give the simplest form. Continuing the process, as seen in the solution, if we identify that 9 and 15 are both divisible by 3, we further simplify the fraction to $-\frac{3}{5}$ to attain the final answer in its simplest form.
Simplified fractions are easier to compare, add, subtract, multiply, or divide when working with other fractions or mixed numbers. They also give a clear and concise representation of the quantity in question.

Negative Exponents Mathematics

Negative exponents introduce an idea that can initially seem counterintuitive: any base raised to a negative exponent is equal to one divided by that base raised to the corresponding positive exponent.
In other words, $a^{-n}$ is equivalent to $\frac{1}{a^n}$ where $a$ is the base and $n$ is a positive integer. It's crucial to differentiate this from a negative base raised to an exponent. A negative base, such as in our exercise ${(-\frac{3}{5})}^3$, affects the sign of the final outcome, particularly when the exponent is odd or even. An odd exponent yields a negative result whereas an even exponent results in a positive value.
Understanding negative exponents is especially important because they frequently appear in scientific calculations, such as in measurements of very small values in scientific notation. Mastering the rules of negative exponents ensures a sound grasp of working with powers of numbers on a more advanced mathematical level.