Problem 25 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{9 b^{3}}-\sqrt{25 b^{3}}+\sqrt{16 b^{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify Each Radical

To simplify the radicals, let's express each term using its prime factors. - For \( \sqrt{9b^3} \): The number inside the square root can be rewritten as \( 9b^3 = (3^2)(b^3)\), and \( \sqrt{3^2 b^3} = 3b^{3/2} \). - For \( \sqrt{25b^3} \): Rewrite it as \( 25b^3 = (5^2)(b^3) \), and \( \sqrt{5^2 b^3} = 5b^{3/2} \). - For \( \sqrt{16b^3} \): Rewrite it as \( 16b^3 = (4^2)(b^3) \), and \( \sqrt{4^2 b^3} = 4b^{3/2} \). After rewriting, we have:\( 3b^{3/2}, 5b^{3/2}, \) and \( 4b^{3/2}\).

Step 2: Arrange and Simplify the Expression

Now that we have simplified the radicals to \( 3b^{3/2} - 5b^{3/2} + 4b^{3/2} \), we can arrange them together to simplify further. Combine like terms by adding or subtracting the coefficients of \( b^{3/2} \): - \( (3 - 5 + 4)b^{3/2} \). - Simplify the coefficients: \( 3 - 5 + 4 = 2 \).So the expression becomes \( 2b^{3/2} \).

Key Concepts

Radicals

Radicals are an essential part of algebra and can sometimes seem tricky, but they become easier as you understand them more deeply. Essentially, a radical is a symbol that represents the root of a number. The most common radical is the square root (\( \sqrt{} \)), which finds a number that, when multiplied by itself, gives the original number.

• For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
• The expression under the radical sign is known as the radicand.

Radicals can be used with variables as well. Consider \( \sqrt{x} \), which means finding a number that squares back to \( x \). In algebra, radicals help in simplifying expressions and solving equations. Keeping an eye on like terms, as we did by taking \( \sqrt{9b^3} - \sqrt{25b^3} + \sqrt{16b^3} \), is crucial for simplification.

Square Roots

Square roots, a type of radical, are specifically used to determine what number, when squared, yields the original number. They are crucial in algebra for solving equations involving squared variables.

• A square root can be simplified by locating perfect squares within the radicand.
• For instance, \( \sqrt{36} \) simplifies to 6 because 36 is a perfect square (\( 6^2 = 36 \)).

To simplify a square root in expressions, consider the factors of the number. Take \( \sqrt{9b^3} \): \( 9b^3 \) becomes \( (3^2)b^3 \) indicating the entire expression can be rewritten using 3 and the square root of \( b^3 \). Such simplifications are useful for easing complex calculations and solving algebraic equations that involve square roots.

Exponents in Algebra

Exponents represent repeated multiplication of a number by itself, enhancing our capacity to express large numbers succinctly. In algebra, they also help in summarizing repeated multiplications involving variables.

• For example, \( b^3 \) means \( b \times b \times b \).
• An exponent in a square root, such as \( b^{3/2} \), indicates the variable is raised to a certain power first and then the root is taken.

Understanding and manipulating exponents is central to simplifying expressions with radicals. For example, converting \( \sqrt{9b^3} \) to \( 3b^{3/2} \) used exponent rules: the square root raised \( b^{3} \) to \( b^{3/2} \). Combining like terms with consistent exponents allows further simplification, as demonstrated when calculating \( 3b^{3/2} - 5b^{3/2} + 4b^{3/2} = 2b^{3/2} \). This step-by-step simplification uses exponents to manage the calculations efficiently.