Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ 3^{2+3} $$

Short answer

Question: Rewrite the expression $$3^{2+3}$$ as a product or a quotient. Answer: $$3^{2+3} = 9 \cdot 27$$

Step-by-step solution

Apply properties of exponents

We are given the expression $$3^{2+3}$$. We can apply the property of exponents which states that $$a^{m+n} = a^m \cdot a^n$$. In this case, our base is 3, and our exponents are 2 and 3. So, we can rewrite the expression as: $$3^{2+3} = 3^2 \cdot 3^3$$

Evaluate the exponents

Now, we need to calculate the value of each exponent. This will give us: $$3^2 = 3 \cdot 3 = 9$$ and $$3^3 = 3 \cdot 3 \cdot 3 = 27$$

Write the final expression

Now, substitute the evaluated values of the exponents back into our expression: $$3^2 \cdot 3^3 = 9 \cdot 27$$ Now, we have successfully rewritten the original expression as a product. So, the final expression is: $$3^{2+3} = 9 \cdot 27$$

Key concepts

Properties of Exponents

When dealing with exponents, understanding their properties simplifies many mathematical expressions. One key property is the product of powers property: for any number with the same base \(a\), raising it to the sum of two exponents, \(m + n\), is equivalent to multiplying two separate powers, \(a^m\) and \(a^n\). This property can be written as \(a^{m+n} = a^m \cdot a^n\).

For example, in the expression \(3^{2+3}\), we can apply this property. Here, the base (3) is raised to the sum of exponents (2 and 3), resulting in \(3^2 \cdot 3^3\). This multiplication mirrors the combining of powers through addition.

Other essential properties include:

• Power of a power: \((a^m)^n = a^{m \cdot n}\)
• Power of a product: \((ab)^n = a^n \cdot b^n\)
• Power of a quotient: \((\frac{a}{b})^n = \frac{a^n}{b^n}\)

Understanding these properties reduces complex expressions and facilitates algebraic manipulation.

Base and Exponent

In any expression with exponents, there are two main components to focus on: the base and the exponent. The base is the number being multiplied by itself, and the exponent tells you how many times to multiply the base.

Consider the expression \(3^{2+3}\). Here, 3 is the base, and the exponent is \(2+3\). This means the base of 3 is repeated in multiplication five times. Learning to identify the base and the exponent is crucial for correctly using the properties of exponents and simplifying expressions.

To break it down further, by understanding each component:

• Base: Shows the number to repeat in multiplication.
• Exponent: Counts how often the base is used in multiplication.

This knowledge allows you to translate and evaluate expressions from a combined form into simpler product or expanded forms.

Evaluating Powers

Evaluating powers involves calculating the value of an expression by performing the necessary multiplications. This requires breaking down the operation indicated by the exponent values.

In the simplified expression \(3^2 \cdot 3^3\), each power must be evaluated separately. Begin by calculating \(3^2\), which means multiplying 3 by itself: \(3 \times 3 = 9\). Next, calculate \(3^3\), which involves multiplying 3 three times: \(3 \times 3 \times 3 = 27\).

Once you have computed these individual powers, the expression becomes straightforward multiplication: \(9 \times 27\). Calculating this product gives you 243, completing the evaluation of the expression.

Tips for efficiently evaluating powers include:

• Break down power expressions into manageable steps.
• Use calculator tools for higher exponent values to save time.
• Double-check each multiplication step to ensure accuracy.

Mastering this skill is vital for solving complex mathematical problems with ease.