Question

\(\sqrt[8]{\frac{n^{18} p^7}{n^2 p^{-1}}}\)

Short answer

The simplified form of \(\sqrt[8]{\frac{n^{18} p^7}{n^2 p^{-1}}}\) is \(n^2 p\)

Step-by-step solution

Simplification

Firstly, simplify the fraction under the root by subtracting the exponents of n and p since it is a division. The rule of exponents states that \(a^{m}/a^{n} = a^{m-n}\). Therefore, it becomes \(\sqrt[8]{n^{18-2} p^{7-(-1)}} = \sqrt[8]{n^{16} p^{8}}\)

Apply Root-Power Rule

Then, apply the Root-Power rule. Root-Power rule states that \(\sqrt[n]{a^m} = a^{m/n}\). So, converting the surd into power, it becomes \((n^{16} p^{8})^{1/8}\)

Simplify Further

Now, apply the rule of power of a power, which states, \((a^m)^n = a^{m*n}\). This enables you to simplify the expression to \(n^{16/8} p^{8/8} = n^{2} p^{1} = n^2 p\)

Key concepts

Root-Power Rule

When dealing with roots and exponents, the root-power rule is incredibly useful. This rule helps you express roots as fractional exponents. Essentially, the rule states that the nth root of a number, raised to the mth power, can be expressed as a single exponent: \[\sqrt[n]{a^m} = a^{\frac{m}{n}}\]This conversion allows for easier calculations and simplifications, especially in algebraic expressions. By translating roots into powers, you can work with regular exponential rules, such as multiplication or division of exponents.
In our exercise, we applied the root-power rule to \((n^{16} p^{8})^{1/8}\), turning them into fractional exponents:

• \(n^{16 \cdot \frac{1}{8}}\)
• \(p^{8 \cdot \frac{1}{8}}\)

Exponent Subtraction

Exponent subtraction is a critical concept to master when working with expressions involving variables raised to powers. This rule comes into play during division operations where the bases are the same. The law states:\[a^{m}/a^{n} = a^{m-n}\]This means you subtract the exponent of the denominator from the exponent of the numerator. In practice, it simplifies the expression significantly. For instance, in our original exercise, we began with \(\frac{n^{18}p^{7}}{n^{2}p^{-1}}\)By applying exponent subtraction:

• The exponents of \(n\) were subtracted: \(n^{18-2} = n^{16}\)
• The exponents of \(p\) were subtracted considering the negative sign: \(p^{7-(-1)} = p^{8}\)

These simplifications pave the way for further manipulation and application of subsequent rules.

Power of a Power Rule

The power of a power rule simplifies expressions where an exponent is raised to another exponent. This rule states:\[(a^m)^n = a^{m \cdot n}\]Applying this rule makes expressions with nested exponents easier to handle, as you simply multiply the exponents together.
In the given exercise, after applying the root-power rule, we dealt with \((n^{\frac{16}{8}} p^{\frac{8}{8}})\)This application transformed it into:

• \(n^{2}\), since \(n^{\frac{16}{8} = n^2}\)
• \(p^{1}\), simplifying to \(p\) since \(p^{\frac{8}{8} = p^1}\)

Hence, the final simplified expression is \(n^2p\). By breaking down each step, utilizing power rules reduces complex expressions into much simpler forms.