Problem 58 - Precalculus Mathematics for Calculus

Question

Simplify the expression. Assume that the letters denote any real numbers. $$\sqrt[3]{a^{2} b} \sqrt[3]{64 a^{4} b}$$

Step-by-Step Solution

Answer

The simplified expression is \(4a^2 b^{2/3}\).

Step 1: Apply the Property of Radicals

Use the property \(\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{xy}\) to combine the two cube roots into a single cube root. The expression becomes: \(\sqrt[3]{a^2 b} \cdot \sqrt[3]{64 a^4 b} = \sqrt[3]{(a^2 b)(64 a^4 b)}\).

Step 2: Simplify Inside the Radicand

Multiply the terms inside the radicand. Combine like terms: \(a^2 \cdot 64 a^4 = 64 a^{6}\) and \(b \cdot b = b^2\). The radicand now becomes \(64 a^6 b^2\), so the expression is \(\sqrt[3]{64 a^6 b^2}\).

Step 3: Simplify the Cube Root Using Exponents

Break down the expression inside the cube root: \(\sqrt[3]{64} = 4\) because \(4^3 = 64\), \(\sqrt[3]{a^6} = a^2\) because \((a^2)^3 = a^6\), and \(\sqrt[3]{b^2} = b^{2/3}\) since \((b^{2/3})^3 = b^2\). Combine these to get: \(4 a^2 b^{2/3}\).

Step 4: Final Expression

The simplified expression is \(4a^2 b^{2/3}\).

Key Concepts

Properties of Radicals

Radicals, commonly known as roots, can be simplified using certain mathematical properties. One key property is that when you multiply two radicals with the same index, you can combine them under a single radical. For example, if you're dealing with cube roots (which have an index of 3), the property is

• \( \sqrt[3]{x} \cdot \sqrt[3]{y} = \sqrt[3]{xy} \)

This tells you that the product of the cube roots of two numbers is the cube root of their product.

In our example, we initially have two cube roots: \( \sqrt[3]{a^2 b} \) and \( \sqrt[3]{64 a^4 b} \). By applying the property mentioned above, you combine these two into a single cube root: \( \sqrt[3]{(a^2 b)(64 a^4 b)} \). This helps to streamline and simplify the expression, making it easier to work with.

Cube Roots

Cube roots are a specific type of radicals where the index number is 3. This means you are looking for the number which, when multiplied by itself three times, gives the original number.
For example, in the expression \( \sqrt[3]{64 a^6 b^2} \), we need to find the cube root of each term.

• \( \sqrt[3]{64} = 4 \): This is because \(4 \times 4 \times 4 = 64 \).
• \( \sqrt[3]{a^6} = a^2 \): Here, you're breaking down the power of 6 into \((a^2)^3\).
• \( \sqrt[3]{b^2} = b^{2/3} \): In this case, the exponent 2 divided by 3 gives you \(b^{2/3}\).

Finding cube roots is simpler when you recognize numbers that are perfect cubes like 64, and understand how to divide exponents by 3.

Exponents

Exponents are a shorthand way to express repeated multiplication. When dealing with cube roots and simplifying expressions, understanding how to work with exponents becomes crucial. Let's break it down:

• When you multiply terms with the same base, you add their exponents. For example, \(a^2 \times a^4 = a^{6} \).
• Taking the cube root of a number or expression means you divide the exponent by 3. For instance, \(a^6\) under a cube root simplifies to \(a^2\), because \(6 \div 3 = 2\).
• For variables with exponents, apply the same rules. The cube root of \(b^2\) links to converting the exponent by dividing it by 3, which results in \(b^{2/3} \).

These rules help you manipulate and simplify expressions, making complex problems much easier to solve.

Real Numbers

In mathematics, we often encounter real numbers. These include all the numbers you can think of except for imaginary numbers, encompassing everything from integers and fractions to radicals and irrational numbers.

When simplifying expressions like \( \sqrt[3]{a^2 b} \sqrt[3]{64 a^4 b} \), it's crucial to understand that the variables involved can represent any real number. This concept allows us to apply the rules and properties of arithmetic and algebra reliably.

• Real numbers include positive and negative figures, as well as zero.
• They follow consistent mathematical rules that allow expressions with radicals, exponents, and coefficients to be simplified uniformly.
• This is why, when simplifying these expressions, we can safely operate under general assumptions based on the properties shared by all real numbers, which provides consistency in mathematical operations.

Understanding real numbers helps ensure that the simplifications we perform adhere to broadly applicable rules, reinforcing solutions that are correct for any real number values of the variables.