Question

Simplify each of the following: (a) \(\frac{a^{11}}{a^{4}}\) (b) \(\left(a^{b}\right)^{4}\)

Short answer

Question: Simplify the following expressions: (a) \(\frac{a^{11}}{a^{4}}\) (b) \(\left(a^{b}\right)^{4}\) Answer: (a) \(a^{7}\) (b) \(a^{4b}\)

Step-by-step solution

Apply the quotient rule in (a)

First, simplify the expression \(\frac{a^{11}}{a^{4}}\). To simplify this expression, we will apply the quotient rule, which states that \(a^{m-n} = \frac{a^m}{a^n}\). Therefore in our case, we have \(a^{11-4} = \frac{a^{11}}{a^{4}}\). So, the simplified expression is \(a^{7}\).

Apply the power of a power rule in (b)

Next, simplify the expression \(\left(a^{b}\right)^{4}\). To simplify this expression, we will apply the power of a power rule, which states that \((a^m)^n=a^{mn}\). Therefore, in our case, we have \(a^{b\cdot 4}=\left(a^b\right)^4\). So, the simplified expression is \(a^{4b}\). The simplified expressions for the given exercise are: (a) \(a^{7}\) (b) \(a^{4b}\)

Key concepts

Quotient Rule

In algebra, the quotient rule helps us simplify expressions involving division of like bases. When you have an expression like \( \frac{a^m}{a^n} \), you use the quotient rule to simplify it. This rule states that you subtract the exponents of the base, resulting in \( a^{m-n} \). Imagine you have \( a^{11} \) divided by \( a^4 \), like in the original problem. Here, both expressions have the same base \( a \). Using the quotient rule, subtract the exponents: 11 (from the numerator) minus 4 (from the denominator). Thus, the simplified result is \( a^7 \).

Why does this work? Essentially, when dividing, you are "canceling out" the equal terms in the numerator and denominator, which leaves you with the remaining expression—essentially the same as subtracting the exponents.

• This rule only works when the bases are the same.
• It simplifies expressions and reduces chances for mistakes.

Remember, algebraic rules are your tools for making expressions manageable and easy to interpret.

Power of a Power Rule

The power of a power rule appears when you have exponents that also have another exponent. For example, \( (a^b)^n \). The rule states that you multiply the exponents: \( a^{b \cdot n} \). It's straightforward!

In the problem \( (a^b)^4 \), this rule simplifies it to \( a^{4b} \). You take the exponent \( b \) of \( a \) and multiply it by 4. By applying this rule, suddenly a complex-looking expression becomes a neat and tidy one.

Why do we multiply? We are raising \( a^b \) to another power, meaning you are using \( a^b \), \( n \) times. Each time you "reuse" \( a^b \), you are effectively adding \( b \) to the exponent, resulting in multiplication.

• You must ensure all variables and bases are consistent first.
• This rule simplifies nested exponential expressions.

This rule is like flattening layers in an expression, resulting in simpler equations.

Exponents Simplification

Simplifying exponents is about turning a complicated expression into an easier one to handle or calculate. When dealing intentionally with exponents, a few key rules help streamline equations without changing the value.

The process involves:

• Applying the quotient rule: Helps with dividing similar bases by reducing the expression through subtraction of exponents.
• Using the power of a power rule:

Power of a Power Rule

This rule simplifies situations where exponents are raised to another power by multiplying the exponents together. In the problem solving practice above, we simplified complex expressions such as \( \frac{a^{11}}{a^4} \) and \( (a^b)^4 \) using these rules.

These rules save time and help reduce the possibility of errors, especially when altering more intricate algebraic expressions. Each rule has its place, but when used correctly, they all turn challenging problems into solvable pieces.