Verify the Equality: 5³ - (4² + 3²) - (√100 + 8²) Step-by-Step

Question

Indicate whether the equality is true or not.

53−(42+32)−(100+82)=53−42−32−100−82 5^3-(4^2+3^2)-(\sqrt{100}+8^2)=5^3-4^2-3^2-\sqrt{100}-8^2

Video Solution

Solution Steps

00:00 Determine whether the equation is correct
00:04 Calculate the exponents and root
00:33 Always solve the parentheses first
00:55 Calculate the exponents and root
01:29 Continue to solve the expression according to the proper order of operations from left to right
01:56 This is the solution

Step-by-Step Solution

In order to determine whether the given equation is correct, we will simplify each of the expressions in its sides separately,

This can be achieved whilst following the order of operations. The order of operations states that exponents precede multiplication and division, which in turn precede addition and subtraction, and that parentheses precede all of the above.

A. Let's start with the expression on the left side of the given equation:

53−(42+32)−(100+82) 5^3-(4^2+3^2)-(\sqrt{100}+8^2)

Begin by simplifying the expressions inside of the parentheses. We'll do this by calculating the numerical value of the terms with exponents (whilst remembering the definition of a root as an exponent, meaning that a root is actually an exponent) Simultaneously we'll calculate the numerical value of the other terms with exponents in the expression:

53−(42+32)−(100+82)=125−(16+9)−(10+64) 5^3-(4^2+3^2)-(\sqrt{100}+8^2) =\\ 125-(16+9)-(10+64)

Finish simplifying the expressions inside of the parentheses, meaning we'll perform the addition operations in them, then we'll perform the remaining subtraction operations:

125−(16+9)−(10+64)=125−25−74=26 125-(16+9)-(10+64) =\\ 125-25-74 =\\ 26

We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

53−(42+32)−(100+82)=125−25−74=26 5^3-(4^2+3^2)-(\sqrt{100}+8^2) =\\ 125-25-74 =\\ 26

B. Let's continue simplifying the expression on the right side of the given equation:

53−42−32−100−82 5^3-4^2-3^2-\sqrt{100}-8^2

Remember the order of operations which states that exponents precede multiplication and division, which precede addition and subtraction, and that parentheses precede all of the above. Note that while this expression has no parentheses, it does have terms with exponents, hence we'll start by calculating the numerical value of the terms with exponents, then we'll perform the subtraction operations:

53−42−32−100−82=125−16−9−10−64=26 5^3-4^2-3^2-\sqrt{100}-8^2= \\ 125-16-9-10-64 =\\ 26

We have finished simplifying the expression on the right side of the given equation, this simplification was brief, so there's no need to summarize it.

Let's now return to the given equation and substitute in its sides the results of simplifying the expressions that were detailed in A and B:

53−(42+32)−(100+82)=53−42−32−100−82↓26=26 5^3-(4^2+3^2)-(\sqrt{100}+8^2)=5^3-4^2-3^2-\sqrt{100}-8^2 \\ \downarrow\\ 26=26

The equation is indeed true, meaning - we obtained a true statement,

Therefore the correct answer is answer A.

Answer

True

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Related Subjects

Order of Operations: Roots Order of Operations: Exponents Order of Operations with Parentheses Order of Operations - Exponents and Roots

Question Types

Powers and Roots: True / false Parentheses in advanced Order of Operations: True / false