Problem 578 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\left(9 p^{\frac{2}{3}}\right)^{\frac{5}{2}}\) (b) \(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\)

Step-by-Step Solution

Answer

(a) 243 p^{\frac{5}{3}}, (b) 81 q^2

Step 1: Simplify the Exponent for Part (a)

For the expression \((9 p^{\frac{2}{3}})^{\frac{5}{2}}\), use the power of a power property \((a^m)^n = a^{m \times n}\). This gives us: \[9^{\frac{5}{2}} \times \left(p^{\frac{2}{3}}\right)^{\frac{5}{2}} = 9^{\frac{5}{2}} \times p^{\frac{2}{3} \times \frac{5}{2}} \]

Step 2: Simplify the Bases for Part (a)

Calculate the exponents: \[9^{\frac{5}{2}} = (3^2)^{\frac{5}{2}} = 3^{2 \times \frac{5}{2}} = 3^5 = 243\] And \[p^{\frac{2}{3} \times \frac{5}{2}} = p^{\frac{10}{6}} = p^{\frac{5}{3}}\] Therefore, \[9^{\frac{5}{2}} \times p^{\frac{5}{3}} = 243 p^{\frac{5}{3}}\]

Step 3: Simplify the Exponent for Part (b)

For the expression \((27 q^{\frac{3}{2}})^{\frac{4}{3}}\), use the power of a power property \((a^m)^n = a^{m \times n}\). This gives us: \[27^{\frac{4}{3}} \times \left( q^{\frac{3}{2}} \right)^{\frac{4}{3}} = 27^{\frac{4}{3}} \times q^{\frac{3}{2} \times \frac{4}{3}} \]

Step 4: Simplify the Bases for Part (b)

Calculate the exponents: \[27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}} = 3^4 = 81\] And \[q^{\frac{3}{2} \times \frac{4}{3}} = q^2\] Therefore, \[27^{\frac{4}{3}} \times q^2 = 81 q^2\]

Key Concepts

Power of a Power Property

To simplify expressions where an exponent is raised to another exponent, the power of a power property is very useful. This property states that \((a^m)^n = a^{m \times n}\). It allows us to combine the two exponents into a single exponent by multiplying them. For example, in the expression \((9 p^{\frac{2}{3}})^{\frac{5}{2}}\), we apply this property:
\[9^{\frac{5}{2}} \times \left(p^{\frac{2}{3}}\right)^{\frac{5}{2}} = 9^{\frac{5}{2}} \times p^{\frac{2}{3} \times \frac{5}{2}}\]
After using the power of a power property, the exponents are now simplified. This crucial first step sets the stage for further simplifications.

Similarly, for \((27 q^{\frac{3}{2}})^{\frac{4}{3}}\), we use the same rule: \[27^{\frac{4}{3}} \times \left( q^{\frac{3}{2}}\right)^{\frac{4}{3}} = 27^{\frac{4}{3}} \times q^{\frac{3}{2} \times \frac{4}{3}}\]\
Mastering this property enables one to tackle complex expressions with ease.

Simplifying Expressions

Once we have applied the power of a power property, the next step is to simplify the resulting expressions. Breaking down each component in \((9 p^{\frac{2}{3}})^{\frac{5}{2}}\) and \((27 q^{\frac{3}{2}})^{\frac{4}{3}}\) helps in simplifying:
\[9^{\frac{5}{2}} = (3^2)^{\frac{5}{2}} = 3^{2 \times \frac{5}{2}} = 3^5 = 243\]
\[p^{\frac{2}{3} \times \frac{5}{2}} = p^{\frac{10}{6}} = p^{\frac{5}{3}}\]
Here, we see the breakdown of \((9 p^{\frac{2}{3}})^{\frac{5}{2}}\). By expressing 9 as \(3^2\), it makes the calculation simpler.

Similarly, we have:
\[27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}} = 3^4 = 81\]
\[q^{\frac{3}{2} \times \frac{4}{3}} = q^2\]
These steps transform the complicated exponents into simpler forms, making the entire expression more manageable.

Rational Exponents

Rational exponents might initially seem tough, but they follow standard exponent rules. A rational exponent means the power is a fraction, such as \(\frac{2}{3}\) or \(\frac{5}{2}\). They can be expressed as radicals:

\[a^{\frac{m}{n}} = \sqrt[n]{a^m}\]
In our example, \(9^{\frac{5}{2}}\) can be perceived as taking the square root (denominator 2) and then raising it to the 5th power. We simplify: \[9^{\frac{5}{2}} = (3^2)^{\frac{5}{2}} = 3^5 = 243\]
By understanding \(a^{\frac{m}{n}}\), we can ease the manipulation of such exponents.

In \((27 q^{\frac{3}{2}})^{\frac{4}{3}}\), \ 27 is represented as \(3^3\), leading to: \27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^4 = 81.\

Breaking complex numbers into simpler bases can make rational exponents much more approachable.

Algebraic Manipulation

Algebraic manipulation involves rewriting and transforming expressions using algebraic rules. It’s essential for solving expressions with exponents. For instance:

In \((9 p^{\frac{2}{3}})^{\frac{5}{2}}\), we start with the power of a power property and simplify: \[9^{\frac{5}{2}} \times (p^{\frac{2}{3}})^{\frac{5}{2}}\]
We factorize and manipulate \ 9 \ as \(3^2\): \[(3^2)^{\frac{5}{2}} = 3^{2 \times \frac{5}{2}} = 3^5 = 243.\]
Another manipulation converts \(p\) exponent: \[p^{\frac{2}{3} \times \frac{5}{2}} = p^{\frac{10}{6}} = p^{\frac{5}{3}}.\]

Similarly,
For \(27 q^{\frac{3}{2}})^{\frac{4}{3}} \, apply the same rules: \[(3^3)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}} = 3^4 = 81\]
and \[q^{\frac{3}{2} \times \frac{4}{3}} = q^2.\]
Effective algebraic manipulation simplifies both \ expressions into more manageable forms: \243 p^{\frac{5}{3}}\) and \81 q^2.\.
Regular practice with these rules enhances fluency in algebra.