Number Comparison: Identifying the Largest Value Among Given Numbers

Question

Select the largest value from among the given options:

Video Solution

Solution Steps

00:00 Choose the largest value
00:03 When multiplying the square root of a number (A) by the square root of another number (B)
00:06 The result equals the square root of their product (A times B)
00:09 Apply this formula to our exercise and calculate the products
00:12 We'll use this method for all expressions in order to find the largest one
00:18 This is the solution

Step-by-Step Solution

In order to determine which of the suggested options has the largest numerical value, we will apply two laws of exponents:

a. Definition of root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

b. Law of exponents for exponents in parentheses (in reverse direction):

anbn=(ab)n a^n\cdot b^n=(a\cdot b)^n

Let's proceed to examine the options a and c (in the answers), starting by converting the square root to exponent notation, using the law of exponents mentioned in a earlier:

2321231216112612 \sqrt{2}\cdot\sqrt{3} \rightarrow 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\\ \sqrt{1}\cdot\sqrt{6} \rightarrow 1^{\frac{1}{2}}\cdot6^{\frac{1}{2}}\\ Due to the fact that both terms in the multiplication have the same exponent, we are able to apply the law of exponents mentioned in b to combine them inside of parentheses, which are subsequently raised to the same exponent. Once completed proceed to calculate the result of the multiplication inside of the parentheses:

212312(23)12=612112612(16)12=612 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}} \rightarrow (2\cdot3)^{\frac{1}{2}}=6^{\frac{1}{2}} \\ 1^{\frac{1}{2}}\cdot6^{\frac{1}{2}}\rightarrow(1\cdot6)^{\frac{1}{2}}=6^{\frac{1}{2}} \\ In the next step, we will return to root notation, again, using the law of exponents mentioned in a (in reverse direction):

6126 6^{\frac{1}{2}}\rightarrow\sqrt{6} \\ We can deduce that the numerical values of options a, b, and c are equal, as seen below:

23212312=612=616112612=612=6 \sqrt{2}\cdot\sqrt{3} \rightarrow 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=6^{\frac{1}{2}}=\sqrt{6}\\ \sqrt{1}\cdot\sqrt{6} \rightarrow 1^{\frac{1}{2}}\cdot6^{\frac{1}{2}}=6^{\frac{1}{2}}=\sqrt{6}\\ Therefore, we need to determine which of these expressions:

6,9 \sqrt{6}, \hspace{6pt}\sqrt{9} has a higher numerical value,

This can be achieved by converting these two values to exponent notation, again, using the law of exponents mentioned in a:

66129912 \sqrt{6}\rightarrow6^{\frac{1}{2}}\\ \sqrt{9}\rightarrow9^{\frac{1}{2}}\\ Note that these two expressions have the same exponent (and their bases are positive), Therefore we can determine their relationship by simply comparing their bases, since it will be identical:

9>6\hspace{4pt} (>0)\\ \downarrow\\ 9^{\frac{1}{2}}>6^{\frac{1}{2}} In other words, we got that:

\sqrt{9}>\sqrt{6}

Therefore, the correct answer is answer d.

Answer

9 \sqrt{9}