Numerical Comparison: Identify the Maximum Value Among Given Numbers

Question

Select the largest value among the given options:

Video Solution

Solution Steps

00:00 Choose the largest value
00:03 When multiplying the root of a number (A) by the root of another number (B)
00:06 The result equals the root of their product (A times B)
00:09 Apply this formula to our exercise and calculate the products
00:12 Apply this method for each expression in order to determine the largest one:
00:25 This is the solution

Step-by-Step Solution

In order to determine which of the following 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 order):

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

Let's start by converting the fourth root in each of the suggested options to exponent notation, using the law of exponents mentioned in a above:

21212112222122122321231224212412 \sqrt{2}\cdot\sqrt{1} \rightarrow 2^{\frac{1}{2}}\cdot1^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{2} \rightarrow 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{3} \rightarrow 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{4} \rightarrow 2^{\frac{1}{2}}\cdot4^{\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 above and combine them together in the multiplication within parentheses, whilst raised to the same exponent. Once completed we can then calculate the result of the multiplication inside of the parentheses:

212212(21)12=212212212(22)12=412212312(23)12=612212412(24)12=812 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}} \rightarrow (2\cdot1)^{\frac{1}{2}}=2^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\rightarrow(2\cdot2)^{\frac{1}{2}}=4^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}} \rightarrow (2\cdot3)^{\frac{1}{2}}=6^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\rightarrow(2\cdot4)^{\frac{1}{2}}=8^{\frac{1}{2}} \\ Let's summarize what we've done so far, as shown below:

21=21222=41223=61224=812 \sqrt{2}\cdot\sqrt{1}=2^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{2}= 4^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{3}= 6^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{4}= 8^{\frac{1}{2}}\\ Now let's note that all the expressions we obtained have the same exponent (they're bases are also positive), therefore we can determine the trend between them using only the trend between their bases, since it's identical to it:

8>6>4>2\hspace{4pt} (>0)\\ \downarrow\\ 8^{\frac{1}{2}}>6^{\frac{1}{2}} >4^{\frac{1}{2}}>2^{\frac{1}{2}}

Therefore the correct answer is answer d.

Answer

24 \sqrt{2}\cdot\sqrt{4}