Unveiling Exponents: Simplifying Complex Expressions

In the realm of mathematics, exponents play a crucial role in simplifying complex expressions and representing extremely large or small numbers. Exponents, also known as powers, provide a concise way to represent repeated multiplication of a base number. Understanding exponents is essential for navigating various mathematical concepts, from basic algebra to advanced calculus.

Defining Exponents

An exponent, denoted by a small raised number to the right of a base number, indicates how many times the base number is multiplied by itself. For instance, in the expression 2³, the base number is 2, and the exponent 3 signifies that 2 is multiplied by itself three times, resulting in 2 x 2 x 2 = 8.

Exponent Rules

Exponents follow specific rules that simplify calculations and manipulations. These rules include:

• Multiplication of Powers with the Same Base: When multiplying terms with the same base, the exponents are added. For example, 2³ x 2⁴ = 2³⁺⁴ = 2⁷ = 128.
• Division of Powers with the Same Base: When dividing terms with the same base, the exponent of the denominator is subtracted from the exponent of the numerator. For instance, 2⁵ / 2² = 2^5-2 = 2³ = 8.
• Power of a Power: When raising a power to another power, the exponents are multiplied. For example, (2³)² = 2^3x2 = 2⁶ = 64.
• Zero Exponent: Any number raised to the power of zero equals one. For instance, 5⁰ = 1.
• Negative Exponents: Negative exponents indicate the reciprocal of the base number raised to the positive exponent. For example, 2^-3 = 1/2³ = 1/8.

Exponent Problem: 6^-2x = -36

Let's apply these exponent rules to solve the given problem: 6^-2x = -36.

Step 1: Isolate the Exponent

To solve for the unknown exponent (-2x), we need to isolate it on one side of the equation.

Step 2: Rewrite as a Positive Exponent

Since we have a negative exponent, let's rewrite the equation with a positive exponent by multiplying both sides by -1.

-6^-2x = -(-36)

Simplifying:

6^2x = 36

Step 3: Solve for x

Now, we need to find the value of x that makes 6^2x equal to 36.

To do this, we can take the logarithm (base 6) of both sides of the equation:

log₆(6^2x) = log₆(36)

Using the exponent rule log_a(a^b) = b, we can simplify:

2x = log₆(36)

Step 4: Calculate x

Using a calculator or logarithm table, we find that:

log₆(36) = 2

Therefore, 2x = 2, which means:

x = 2/2

Final Answer:

x = 1

Conclusion

In conclusion, exponents provide a powerful tool for simplifying mathematical expressions and calculations. Understanding exponent rules and their application is crucial for solving various mathematical problems, including those involving negative exponents. By following the steps outlined in this blog, you can confidently solve exponent problems and expand your mathematical knowledge.