Problem 90 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify. Write your answer as a power or as an expression containing powers. $$\left(3^{6}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified form of \((3^{6})^{3}\) is \(3^{18}\)

Step 1: Identify the base and the exponents

The expression \((3^{6})^{3}\) has a base of 3. The first exponent is 6 and then this result is raised to the power of 3.

Step 2: Apply the rule of exponents

According to the rule of exponents, \((a^{m})^{n} = a^{mn}\), the exponents should be multiplied when a power is raised to another power. Here, you need to multiply 6 and 3 to get the final exponent.

Step 3: Simplify the expression

Multiply the exponents. \(6 \times 3 = 18\). So, \((3^{6})^{3} = 3^{18}\)

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For example, in the expression \(3^6\), the number 3 is the base and 6 is the exponent. This tells us that 3 is multiplied by itself 6 times, or \(3 \times 3 \times 3 \times 3 \times 3 \times 3\).
Exponents make it simpler to write and work with large numbers. Instead of writing out long multiplication, we use exponents to communicate the same idea with fewer symbols.
Additionally, exponents can be positive, negative, or even fractional. Each type of exponent has a different implication on how the base is used to calculate the final result. For simplicity:

• A positive exponent means repeated multiplication.
• A negative exponent indicates the reciprocal (or inverse) of repeated multiplication.
• A fractional exponent relates to roots of the number.

The Concept of Powers

Powers are closely related to exponents and can be seen as the full expression involving a base and an exponent. For example, in \(3^6\), the entire expression is referred to as a power. This includes both the base (3) and the exponent (6).
Powers are helpful for simplifying calculations and understanding scaling in mathematics. When you see a squared value, such as \(x^2\), or a cubed value, \(y^3\), you're working with powers. Powers allow us to handle very large or very precise numbers easily by indicating how many times a base number should be multiplied by itself.
Using powers, we can describe and solve various mathematical problems effectively, ranging from simple arithmetic to complex scientific calculations.

Laws of Exponents

The laws of exponents are rules that guide how exponents interact with one another. These laws make operations like multiplication, division, and raising powers to powers easier to manage. Here's a brief overview of some important laws:

• Product of Powers: When multiplying like bases, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing like bases, subtract the exponents. \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: To raise a power to another power, multiply the exponents. \((a^m)^n = a^{m \times n}\).
• Power of a Product: To find the power of a product, apply the exponent to each factor. \((ab)^m = a^m \times b^m\).

These rules simplify working with expressions that have exponents. In the given exercise, we specifically used the "Power of a Power" rule to simplify \((3^6)^3\) into \(3^{18}\). Understanding these laws is crucial for handling more complex mathematical tasks efficiently.