Question

Write the expression in the form \(x^{n}\), assuming \(x \neq 0\) $$ \frac{x \cdot x^{6}}{\left(x^{3}\right)^{2}} $$

Short answer

Question: Simplify the given expression to the form \(x^n\): \(\frac{x \cdot x^{6}}{\left(x^{3}\right)^{2}}\) Answer: The simplified expression is \(x\).

Step-by-step solution

Rewrite the expression using the properties of exponents

The given expression is: $$ \frac{x \cdot x^{6}}{\left(x^{3}\right)^{2}} $$ First, let's use the power rule of exponents \((a^m)^n = a^{mn}\) to rewrite the denominator: $$ \left(x^{3}\right)^{2} = x^{3 \cdot 2} = x^6 $$ This gives us: $$ \frac{x \cdot x^{6}}{x^6} $$

Simplify the expression using the properties of exponents

Now, we can use the multiplication rule of exponents \(a^m \cdot a^n = a^{m+n}\) to combine the terms in the numerator: $$ x \cdot x^{6} = x^{1+6} = x^7 $$ This gives us the expression: $$ \frac{x^7}{x^6} $$ Finally, we can use the division rule of exponents \(a^m \div a^n = a^{m-n}\) to simplify the fraction: $$ \frac{x^7}{x^6} = x^{7-6} = x^1 $$ So the simplified expression is \(x^1\), which can also be written simply as \(x\).

Key concepts

Power Rule of Exponents

Understanding the power rule of exponents is crucial when dealing with expressions involving powers raised to other powers. This rule states that when you have a power raised to another power, you multiply the exponents. algebraically, it is represented as

• \((a^m)^n = a^{m \cdot n}\)

For example, in our exercise, the denominator \((x^3)^2\) is a perfect scenario to use this rule. By applying the power rule, we convert it from \((x^3)^2\) to \(x^{3 \cdot 2}\), which simplifies to \(x^6\). This shows how efficiently we can simplify expressions by understanding and applying the correct property of exponents. Remember, multiplying the exponents only applies when raising a power to another power.

Multiplication Rule of Exponents

The multiplication rule of exponents helps us simplify terms with the same base multiplied together. According to this rule, when you multiply numbers with the same base, you add the exponents.

• \(a^m \cdot a^n = a^{m+n}\)

In the exercise above, the multiplication of \(x\) and \(x^6\) in the numerator illustrates this rule. By applying it, we combine these terms into one term: \(x^{1+6}\), which becomes \(x^7\). It's important to notice that even if no exponent is visible, a base like \(x\) can be seen as \(x^1\). This ensures we don't miss out any part of the expression when calculating the new exponent. This rule makes it simpler to work with expressions where multiple terms share the same base.

Division Rule of Exponents

The division rule of exponents is essential for simplifying fractions where the bases are the same. When dividing terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

• \(a^m \div a^n = a^{m-n}\)

In our given exercise, we divide \(x^7\) by \(x^6\), following the division rule. The exponents subtract, resulting in \(x^{7-6}\) which equals \(x^1\). In fact, it simplifies further to just \(x\) because any term raised to the power of one is the term itself. Using this rule allows for elegant simplification of expressions that initially look more complicated due to their powers. Recognizing these patterns and rules helps to streamline calculations and fosters deeper understanding.