Mat1082639_en 10 ÷ (10 ÷ 10 ÷ 10) − (−4)² − (−5²) = ?

 

VERSION 1 — DETAILED EXPLANATION

Problem:  10 ÷ (10 ÷ 10 ÷ 10) − (−4)² − (−5²) = ?

We use the PEMDAS rule:

P = Parentheses
E = Exponents
M/D = Multiplication and Division
A/S = Addition and Subtraction

IMPORTANT RULE:

Multiplication and Division have the SAME priority, so they are calculated from LEFT to RIGHT.

Addition and Subtraction also have the SAME priority, so they are calculated from LEFT to RIGHT.

There is also an important difference between:   (−5)² and −5²

They are NOT the same:  (−5)² = 25

But:  −5² = −25

Because the exponent is applied before the negative sign when there are no parentheses around −5.

STEP 1 — SOLVE THE PARENTHESES

The expression inside the parentheses is:  10 ÷ 10 ÷ 10

Division has the same priority throughout, so calculate from LEFT to RIGHT.

First:  10 ÷ 10 = 1

Then:  1 ÷ 10 = 1/10

Therefore:

10 ÷ 10 ÷ 10
= 1 ÷ 10
= 1/10

The original expression becomes:   10 ÷ 1/10 − (−4)² − (−5²)

STEP 2 — SOLVE THE DIVISION

Now calculate:   10 ÷ 1/10

Dividing by a fraction means multiplying by its reciprocal:

10 ÷ 1/10
= 10 × 10
= 100

Therefore:   100 − (−4)² − (−5²)

STEP 3 — SOLVE THE EXPONENTS

First:  (−4)²

Because −4 is inside parentheses:

(−4)²
= (−4) × (−4)
= 16

Now:  −5²

The exponent applies to 5 first:

−5²
= −(5²)
= −25

Therefore:  (−5²) = −25

The expression becomes:  100 − 16 − (−25)

STEP 4 — SUBTRACTION OF A NEGATIVE NUMBER

We have:  100 − 16 − (−25)

Subtracting a negative number is equivalent to adding a positive number:   −(−25) = +25

Therefore:   100 − 16 + 25

STEP 5 — ADDITION AND SUBTRACTION FROM LEFT TO RIGHT

First:   100 − 16 = 84

Then:  84 + 25 = 109

 

FINAL ANSWER:

10 ÷ (10 ÷ 10 ÷ 10) − (−4)² − (−5²) = ?
= 10 ÷ (1/10) − 16 − (−25)
= 100 − 16 + 25
= 84 + 25
= 109

ANSWER = 109

 

VERSION 2 — SUMMARY WITHOUT EXPLANATION
ENGLISH

10 ÷ (10 ÷ 10 ÷ 10) − (−4)² − (−5²) = ?
= 10 ÷ (1/10) − 16 − (−25)
= 100 − 16 + 25
= 84 + 25
= 109

ANSWER = 109

 

IMPORTANT PEMDAS NOTE:

The most important part of this problem is:  (−5²)

This is NOT the same as:  (−5)²

Compare:  (−5)² = 25

while:  −5² = −25

Therefore: 

−(−5²)
= −(−25)
= +25

This is why the final answer is:  109

Problem Author: Aris Nurdiana (Public Platform: Facebook)

Step-by-Step Solution: Akur Atan (Accuracy)

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