Problem 11 - Advanced Calculus

Question

Show that for a positive number \(x\) and integers \(m\) and \(n,\) with \(n\) positive, $$ \left(x^{1 / n}\right)^{m}=\left(x^{m}\right)^{1 / n} $$

Step-by-Step Solution

Answer

By applying the exponent rules to both sides, we show that \((x^{1/n})^m = x^{m/n}\) and \((x^m)^{1/n} = x^{m/n}\), thus proving that both expressions are equal.

Step 1: Understand The Property of Exponents

Recognize that the property of exponents states that for any positive number a, and integers p and q, the expression \((a^{p})^{q} = a^{p \cdot q}\) holds. This can be used to simplify both sides of the given equation.

Step 2: Rewrite Left-Hand Side Using Exponent Rules

Apply the exponent rule to the left-hand side of the equation: \((x^{1/n})^m\) can be rewritten as \(x^{(1/n) \cdot m}\), which simplifies the expression to \(x^{m/n}\).

Step 3: Rewrite Right-Hand Side Using Exponent Rules

Apply the exponent rule to the right-hand side of the equation: \((x^{m})^{1/n}\) can be rewritten as \(x^{m \cdot (1/n)}\), which simplifies the expression to \(x^{m/n}\).

Step 4: Compare Simplified Sides

Observe that after simplifying both sides using the exponent rules, we have \(x^{m/n}\) on the left-hand side and \(x^{m/n}\) on the right-hand side, showing that they are indeed equal.

Key Concepts

Understanding Positive Number Exponents

Positive number exponents are the bread and butter of algebra. When you raise a number to a positive exponent, you're essentially multiplying that number by itself a certain number of times. For example, if we take the number 3 and raise it to the power of 4 (written as \(3^4\)), we multiply 3 by itself 4 times: \(3 \times 3 \times 3 \times 3\), which equals 81.

But what happens when we have fractions in the exponent? This might seem tricky at first, but it's all based on the idea of roots. If you have an expression \(x^{1/n}\), it's the same as taking the nth root of x. So \(9^{1/2}\) is the square root of 9, which is 3. This concept is essential when working with different types of exponential expressions and sets the stage for simplifying complex equations.

Demystifying Integer Exponents

Integer exponents might seem like they only deal with whole numbers, but they can take on negative values and even zero. When you raise a number to a negative exponent, such as \(x^{-n}\), you are actually taking the reciprocal of the number raised to the positive exponent: \(\frac{1}{x^n}\). For example, \(5^{-2}\) is the same as \(\frac{1}{5^2}\), which simplifies to \(\frac{1}{25}\).

Zero Exponents

Anything raised to the power of zero, except zero itself, is always 1. So, \(x^0 = 1\), regardless of the value of x. This is because any number divided by itself equals 1, and raising a number to the power of zero is somewhat equivalent to dividing it by itself an infinite amount of times.

By mastering integer exponents, you equip yourself with a powerful tool for algebra and beyond, making it easier to tackle equations and simplify expressions. It's essential to remember that integer exponents provide a shorthand for describing repeated multiplication or division.

Simplifying Expressions with Exponents

Simplifying expressions with exponents is a critical skill that allows you to transform complex looking expressions into simpler forms. This can make it easier to solve equations, compare expressions, or evaluate numbers.

Combining Like Terms

When simplifying, always look for like terms—terms that have the same base and exponent. For instance, \(x^3 + 2x^3\) simplifies to \(3x^3\) because both terms have the base 'x' raised to the power of 3.

Distributive Property

The distributive property can help when you have exponents inside parentheses with a coefficient outside, like in \(3(x^2 + x^3)\). You'd multiply the coefficient by each term inside the parentheses: \(3x^2 + 3x^3\).

Incorporating the rules of exponents, such as those illustrated in the textbook solution, can further simplify expressions. This involves adding, subtracting, or even dividing exponents depending on their bases and whether they're multiplied or divided. Simplifying expressions may seem daunting at first, but by taking it step by step and applying the rules consistently, you'll soon see a clear and more manageable path through the thicket of exponents.