%5E-1%2F2-.%24.jpeg "$(2^4 Times 5^6)^ 1/2 .$")
Simplifying Expressions with Exponents
This article will guide you through simplifying the expression $(2^4 \times 5^6)^{1/2}$.
Understanding the Properties of Exponents
Before we dive into the simplification, let's review some key properties of exponents:
- Product of Powers: When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$
- Power of a Power: When raising a power to another power, multiply the exponents: $(a^m)^n = a^{m \times n}$
- Fractional Exponents: A fractional exponent indicates a root. For example, $a^{1/n} = \sqrt[n]{a}$
Simplifying the Expression
-
Apply the Power of a Power Property: We begin by applying the power of a power property to the expression inside the parentheses:
$(2^4 \times 5^6)^{1/2} = 2^{4 \times (1/2)} \times 5^{6 \times (1/2)}$
-
Simplify the Exponents: Next, simplify the exponents:
$2^{4 \times (1/2)} \times 5^{6 \times (1/2)} = 2^2 \times 5^3$
-
Calculate the Powers: Finally, calculate the remaining powers:
$2^2 \times 5^3 = 4 \times 125 = 500$
Conclusion
Therefore, the simplified form of $(2^4 \times 5^6)^{1/2}$ is 500. By understanding the properties of exponents and applying them systematically, we can simplify complex expressions into their simplest form.