Question

Write with a single exponent. $$ \frac{2^{a} 3^{a}}{6^{b}} $$

Short answer

Question: Simplify the expression $\frac{2^{a} 3^{a}}{6^{b}}$. Answer: $6^{\frac{a}{b} - b}$

Step-by-step solution

Rewrite the expression as a product of powers with the same base

In order to simplify the expression, we should rewrite it to have a single base. Since 2 and 3 are factors of 6, it helps to rewrite the expression as powers of 6: $$ \frac{2^{a} 3^{a}}{6^{b}} = \frac{6^{\frac{a}{b}}}{6^{b}} $$

Use the properties of exponents to rewrite the expression with a single exponent

Now that the expression is written with the same base, we can use the properties of exponents to simplify it further. Specifically, we can subtract the exponent in the denominator from the exponent in the numerator: $$ \frac{6^{\frac{a}{b}}}{6^{b}} = 6^{\frac{a}{b} - b} $$ This expression is now written with a single exponent, as required.

Key concepts

Properties of Exponents

When working with exponents, there are several rules and properties that can help simplify expressions. These rules allow us to manipulate the expressions in a consistent and efficient manner. Here are some key properties you should keep in mind:

• Product of Powers: When multiplying two expressions with the same base, you can add the exponents: \( x^m \cdot x^n = x^{m+n} \).
• Quotient of Powers: When dividing two expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator: \( \frac{x^m}{x^n} = x^{m-n} \).
• Power of a Power: When taking a power to another power, multiply the exponents: \( (x^m)^n = x^{m \cdot n} \).

In our exercise, the key property used is the quotient of powers. We see the expression has the same base in the numerator and the denominator which allows us to subtract the exponents. This property helps turn the expression into a simpler form.

Simplifying Expressions

Simplifying expressions is about making them as concise and clear as possible without changing their value or outcome. In mathematics, it's often beneficial to have simpler forms to work with, especially when solving equations or performing further calculations.

To simplify exponents, always try to:

• Combine like terms, in this case expressions with the same base.
• Utilize properties of exponents such as the product and quotient rules discussed earlier.
• Reduce fractions in expressions when possible.

The goal is to have a streamlined form that represents the original statement accurately. In this exercise, simplifying meant rewriting \( \frac{2^{a} 3^{a}}{6^{b}} \) into an expression with a single exponent, which made further manipulation easier.

Powers and Roots

Understanding powers and roots is crucial as they are foundational concepts in algebra and beyond. In simple terms, the power of a number refers to how many times a number, called the base, is multiplied by itself. For example, \( x^3 \) means \( x \times x \times x \).

Roots, on the other hand, are the inverse operation. A square root, for instance, asks which number multiplied by itself gives the original number. In mathematical terms, \( \sqrt{x} \), for a non-negative \( x \), finds a number \( y \) such that \( y^2 = x \).

Powers and roots are interconnected, and understanding this relationship is vital for simplifying expressions that involve them. In our exercise, we did not directly deal with roots, but understanding powers is essential as it helps relate and simplify the terms efficiently. The key takeaway is recognizing the base and managing exponents, laying a strong groundwork for any operations involving powers.