Question

Simplify the expressions, assuming all variables are positive. $$ \sqrt{48 a^{3} b^{7}} $$

Short answer

Answer: $$4\sqrt{3} \cdot a^{\frac{3}{2}} \cdot b^{\frac{7}{2}}$$

Step-by-step solution

Simplify the constants under the square root

We need to simplify the constant 48 under the square root. To do this, we can find the largest perfect square that can be divided into 48 and then take the square root of that perfect square. 48 = 16 * 3, where 16 is the largest perfect square that can be divided into 48. So, $$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \sqrt{3} = 4\sqrt{3}$$

Simplify the variables under the square root

Now let's simplify the variables under the square root. For a^3, we can apply the rule of properties of the square root which states: $$\sqrt{x^n} = x^{\frac{n}{2}}$$ $$\sqrt{a^3} = a^{\frac{3}{2}}$$ For b^7, we can do the same: $$\sqrt{b^7} = b^{\frac{7}{2}}$$

Combine the simplified constants and variables

Now that we have simplified the constants and variables, we can combine them. $$ \sqrt{48 a^{3} b^{7}} = 4\sqrt{3} \cdot a^{\frac{3}{2}} \cdot b^{\frac{7}{2}} $$

Key concepts

Perfect Square

When dealing with radical expressions, identifying perfect squares can simplify the process remarkably. A perfect square is a number that is the square of an integer. For example:

• The number 1 is a perfect square because it equals 1 squared (1×1).
• Similarly, 4 is a perfect square since it is 2 squared (2×2).
• 16, 25, 36 are all perfect squares (4×4, 5×5, 6×6 respectively).

Perfect squares are particularly useful in simplifying square root expressions because they can be "pulled out" of the radical. For instance, in the expression \( \sqrt{48} \), we can break 48 into 16 and 3 (where 16 is a perfect square). The square root of 16 is simply 4, so \( \sqrt{48} \) becomes \( 4 \sqrt{3} \). This extraction helps simplify expressions, making them easier to manage.

Properties of Square Roots

Square root properties are essential for simplifying expressions. The primary property says:

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) (product property)
• If \( x \) is squared to give \( x^2 \), then \( \sqrt{x^2} = x \).
• For variables: \( \sqrt{x^n} = x^{n/2} \).

Let's apply these rules. In our expression \( \sqrt{48 a^3 b^7} \), we used:- The product property of square roots to split 48 as \( 16 \times 3 \).- Then, for \( a^3 \), the property lets us handle it as \( a^{3/2} \), giving a simplified form. - The same simplification applies to \( b^7 \): \( \sqrt{b^7} = b^{7/2} \).Using these properties helps to tackle complex expressions by breaking them down.

Exponents

Exponents play a significant role in manipulating and simplifying expressions under a square root. Understanding exponent rules, especially when combined with roots, becomes crucial. Here are some basics:

• An exponent denotes how many times to multiply a number by itself: for example, \( a^3 = a \times a \times a \).
• The quotient rule \( a^{m-n} = \frac{a^m}{a^n} \) can be useful to simplify expressions.
• When dealing with square roots: \( \sqrt{x^n} = x^{n/2} \). This tells us that each power under a square root is divided by 2.

In the example \( \sqrt{a^3} \), \( a \) raised to the 3 is simplified by taking half the power under the square root, resulting in \( a^{3/2} \). This approach allows us to manage terms under square roots efficiently, leading to clear and simplified expressions. Understanding these rules empowers us to tackle even more complex algebraic problems.