Question

Can you tell me the answers to these in simplest form?1. -36 ÷ (-9)2. 8² ÷ 2⁴3. -3⁴ over 3³ 4. -4(-8) ÷ (-2)5. 3⁰ + 1⁰6. -2² + 3²7. 3² over 3^negative²8. 36 ÷ (18 ÷ 2)9. y⁶× y³10. Determine whether the following is positive, negative, or zero: -3(-2)³(4)(-5)⁴ over 3²(2)(-6)⁴11. Order the following numbers from least to greatest by using < : 2³; (-2)²; 2-³; -2^ negative ²

Answer 1

1. (-36)÷(-9)

When you multiply or divide two negative numbers, the negative signs cancel each other and the result is always positive.

This division can also be expressed as a fraction:

`(-36)/(-9)=(36)/(9)\text{use common factor 9}(36/9)/(9/9)=(4)/(1)=4`

2. 8²÷2⁴

Best is to solve the exponents first

8²=8*8=64

2⁴=2*2*2*2=16

64÷16=4

3.

`(-3^4)/(3^3)`

-3⁴= 3⁴= 81

3³= 27

81/27=3

When dividing exponents with the same base, you can also subtract the exponents and then solve the calculation:

3⁴÷3³=3⁴⁻³=3¹=3

4.

[(-4)(-8)÷(-2)]

In this example, you have to follow the order of the operations from left to right because there are no ( ) to give you another order.

So first multiply (-4)(-8)= 32

Then divide by -2

32÷(-2)= -16

5. 3⁰+1⁰

As a rule, any number to the power 0 equals 1, always, regardless which number it is, so

3⁰=1

1⁰=1

3⁰+1⁰=1+1=2

6. (-2)²+3²

(-2)²=2²=4 (because the exponent is even, this means when the numbers are multiplied, the negative signs cancel each other)

(-2)²+3²= 4 + 9= 13

7.

`(3^2)/(3^(-2))`

The negative exponent indicates that the value is inverted, for example 3⁻²=1/(3²)= 1/9

To solve this you can do the same operation as in item 3. because both exponents have the same base. Remember that when you divide two exponents with the same base, you have to subtract the exponent of the numerator minus the exponent of the denominator:

3²/3⁻²=3²⁻(⁻²) →is a dounble negative so they cancel each other

=3²⁺²=3⁴=81

If you divide 9 by 1/9 you'll reach the same result.

8. 36÷(18÷2)

In this example one of the operations in between ( ), as a rule, you always solve the operation inside the parenthesis first and then the others. So the first step is to divide 18 by 2

18÷2=9

Now you have to divide 36 by the result of the first operation:

36÷9=4

9.

y⁶*y³

Following the rules of exponentials, when you multiply two exponentials with the same base, in this case the unknown y, you have to add both exponents:

y⁶*y³=y⁶⁺³=y⁹

10.

[-3(-2)³(4)(-5)⁴]/[3²(2)(-6)⁴]

Lets start with the numerator

-3(-2)³(4)(-5)⁴

(-2)³: the exponent is uneven, so as a rule, you can say that the result will be negative. (-2)³=-8

(-5)⁴: the exponent is even, so the result will be positive. (-5)⁴=625

(-3)*(-8)*4*625

Solve the operations from left to right, the fist multiplication is between two negative numbers so the signs cancel each other, leading to a positive result:

(-3)*(-8)=24

24*4=96

96*625=60000

Denominator

3²(2)(-6)⁴

(-6)⁴: is the only negative number and is powered by 4, the exponent is an even number so the result will be positive. (-6)⁴=1296

The general result of the denominator will be a positive number:

3²(2)(-6)⁴=9*2*1296=23328

The division will be between two positive numbers and its result will also be positive:

60000/23328=2.57

11.

• 2³ will be a positive result 2³=8

,

• (-2)² will result in a positive value (-2)²=4

,

• 2⁻³ the exponent is negative, this means the value will be inverted 2⁻³=1/2³=1/8

,

• -2⁻² The exponent is negative, which means the value is inverted, -2⁻²= -1/2² and is even which leads to a positive result. -1/2²=1/4

Now that the exponents are solved you have order them from least to greatest:

2⁻³<(-2)⁻²<(-2)²<2³

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