We included **HMH Into Math Grade 8 Answer Key PDF** **Module 12 Review **to make students experts in learning maths.

## HMH Into Math Grade 8 Module 12 Review Answer Key

**Vocabulary**

**In Problems 1-3, complete each sentence with the correct operation to explain the properties of exponents.**

Question 1.

To raise a power to a power, keep the base the same and _____________ the exponents.

Answer:

Multiply,

Explanation:

The power rule for exponent is to raise a power to a power, keep the base

the same and multiply the exponents.

Example: (2^{3})^{5} = 2^{15}.

Question 2.

To divide two powers with the same base, keep the base the same and ____________ the exponents.

Answer:

Subtract,

Explanation:

To divide two powers with the same base keep the base the same and subtract the exponents.

Question 3.

To multiply two powers with the same base, keep the base the same and ____________ the exponents.

Answer:

Add,

Explanation:

To multiply two powers with the same base keep the base the same and add the exponents.

Question 4.

What is the difference between the standard form of a number and the number written in scientific notation?

Answer:

Standard notation is the normal way of writing numbers, Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form (such as 10^{3}, 10^{-3}, 10^{12} etc),

Explanation:

Standard notation is the normal way of writing numbers. Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a

way of expressing numbers that are too big or too small to be conveniently written in decimal form.

To change a number from scientific notation to standard form, move the decimal point to the left

(if the exponent of ten is a negative number) or to the right (if the exponent is positive).

We should move the point as many times as the exponent indicates.

An example of scientific notation is 1.3 ×10^{6 }which is just a different way of expressing the standard notation of the number 1,300,000.

**Concepts and Skills**

Question 5.

**Use Tools** Write an expression with a single exponent that is equivalent to 8^{-4} • (8^{2})^{4}. State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.

Answer:

Expression: 8^{4},

Explaantion:

Asked to write an expression with a single exponent that is equivalent to 8^{-4} • (8^{2})^{4} used rules of exponents as the power rule for exponent is to raise a power to a power,

keep the base the same and multiply the exponents.

So (8^{2})^{4 }= (8^{2×4})= (8^{8}), Now when two exponential terms with the same base are multiplied their powers are added while the base remains the same. So 8^{-4} • (8^{8}) = 8 ^{-4+8} = 8^{4}.

Question 6.

Compare each expression to 6^{6}.

Answer:

Explanation:

Given expressions to campare with 6^{6 }, So 1. (6^{-1} • 6^{4})^{2}, As bases are same powers are added,

So (6^{-1+4})^{2} = (6^{3})^{2}, As exponent is to raise a power to a power we keep the base the same and multiply the exponents.

So (6^{3})^{2 }= 6^{3X2 }= 6^{6}, So (6^{-1} • 6^{4})^{2 }= 6^{6}, 2.(6^{8})/(6^{2})^{2 }, As exponent is to raise a power to a power we

keep the base the same and multiply the exponents. So (6^{8})/(6^{2X2})^{ }= (6^{8})/(6^{4}) , As to divide two powers with the same base, keep the base the same and subtract the exponents.

So (6^{8 – 4}) = 6^{4 },So as 6^{4 }< 6^{6}, 3. (6^{2}/6^{0})^{4 }, As to divide two powers with the same base,

keep the base the same and subtract the exponents.

So (6^{2-0})^{4 },= (6^{2})^{4 }, Now when exponent is to raise a power to a power we keep the base the same and

multiply the exponents so (6^{2})^{4 }= 6^{2x}^{4 }= 6^{8} > 6^{6}, Completed teh table as shown above.

Question 7.

Select all the expressions equivalent to \(\frac{9^{3} \cdot 9^{5}}{9^{2}}\).

(A) 3^{8}

(B) 3^{12}

(C) 9^{6}

(D) 9^{4}

(E) 27^{2}

(F) 27^{4}

Answer:

(B) 3^{12}, (C) 9^{6 }and (F) 27^{4},

Explanation:

Given \(\frac{9^{3} \cdot 9^{5}}{9^{2}}\) when bases are same powers are added so (9^{3 + 5}) / 9^{2 }= 9^{8}/ 9^{2 }, now as to divide two powers with the same base keep the base the same and subtract the exponents. So 9^{8}/ 9^{2 }= 9^{8-2 }= 9^{6}, 9^{6 }is equivalent to (3 X 3)^{6 }= (3^{1 }X 3^{1})^{6 }= (3^{1+1})^{6 }= 3^{2X6 }= 3^{12 }and 9^{6 }is equivalent to (3 X 3 X 3 ) X (3 X 3 X 3) X (3 X 3 X 3) X (3 X 3 x 3) as bases are same powers are added so it is (27)^{4}, therefore the expressions equivalent to \(\frac{9^{3} \cdot 9^{5}}{9^{2}}\) are

bits (B) 3^{12}, (C) 9^{6 }and (F) 27^{4}.

Question 8.

What are possible values for a and b in the equation \(\frac{4^{a}}{4^{b}}\) = 4^{-1}? What must be true about the values of a and b?

a = ____________

b = ____________

Answer:

a = 0 and b = 1,

Explanation:

The possible values for a and b in the equation \(\frac{4^{a}}{4^{b}}\) = 4^{-1}.

It must be true about the values of a and b are \(\frac{4^{a}}{4^{b}}\) = \(\frac{1}{4}\), as we know any number to power of zero(0) is 1 and as 4^{b }= 4 it means that any number if it has power 1 then its base is same so 4^{1}= 4, therefore \(\frac{4^{a}}{4^{b}}\) =

\(\frac{4^{0}}{4^{1}}\), therefore a = 0 and b = 1.

Question 9.

The diameter of Earth is about 1 × 10^{4} kilometers, and the diameter of a basketball is about 2 × 10^{-4} kilometer. About how many times as great is the diameter of Earth as the diameter of a basketball?

(A) 5000 times as great

(B) 50,000 times as great

(C) 50,000,000 times as great

(D) 500,000,000 times as great

Answer:

(C) 50,000,000 times as great,

Explanation:

Given the diameter of Earth is about 1 × 10^{4} kilometers and the diameter of a basketball is about 2 × 10^{-4} kilometer.

Number of times as great is the diameter of Earth as the diameter of a basketball is 1 × 10^{4} kilometer/ 2 × 10^{-4} kilometer.

Question 10.

A bee hummingbird has a mass of 0.0023 kilograms. What is the mass of a bee hummingbird written in scientific notation?

____________ kilogram

Answer:

The mass of a bee hummingbird written in scientific notation is 2.3 × 10^{-3},

Explanation:

Scientific notation is a way of expressing numbers that are too large or too small, written in the form of power of 10. So 0.0023 kilograms to scientific notation with steps.

All the numbers in scientific notation written in the form a × 10^{b}, where b is an integer and

the coefficient a is a non-zero real number between 1 and 10 in absolute value.

To convert 0.0023 into scientific notation also known as standard form, follow these steps:

Move the decimal 3 times to right in the number so that the resulting number,

m = 2.3, is greater than or equal to 1 but less than 10,

Since we moved the decimal to the right the exponent n is negative n = -3, Write in the scientific notation form, m × 10^{n }= 2.3 × 10^{-3}. Therefore, 0.0023 in scientific notation is 2.3 × 10^{-3} and

It has 2 significant figures. In scientific e-notation it is written as 2.3e-3.

Therefore the mass of a bee hummingbird written in scientific notation is 2.3 × 10^{-3}.

Question 11.

Naomi says that 9.8 × 10^{5} is greater than 3.2 × 10^{6}. Is Naomi correct? Explain your reasoning.

Answer:t

No, Naomi is incorrect,

Explanation:

Given to find Naomi says that 9.8 × 10^{5} is greater than 3.2 × 10^{6}.

Is Naomi correct, No Naomi is incorrect as 9.8 × 10^{5} is not greater than 3.2 × 10^{6}. because if we see 3.2 X 10^{6 }= 32 × 10^{5}, now if compare both sides we have 10^{5} and now if we compare

9.8 and 32, 32 is greater than 9.8, So 3.2 × 10^{6} is greater than 9.8 × 10^{5}, therefore Naomi is incorrect.

Question 12.

Which expression is equivalent to \(\frac{\left(8 \times 10^{3}\right)+\left(4 \times 10^{3}\right)}{\left(3 \times 10^{-2}\right)}\)?

(A) 4 × 10^{1}

(B) 4 × 10^{4}

(C) 4 × 10^{5}

(D) 4 × 10^{8}

Answer:

(D) 4 × 10^{8},

Explanation:

Equivalent expression to \(\frac{\left(8 \times 10^{3}\right)+\left(4 \times 10^{3}\right)}{\left(3 \times 10^{-2}\right)}\) is as bases are same powers are added so it is 12 X 10^{3+3}/3 X 10^{-2 }=

12 X 10^{6}/3 X 10^{-2 }to divide two powers with the same base, keep the base the same and subtract the exponents so 4 X 10^{6+2 }= 4 X 10^{8 }which matches with bit (D).

Question 13.

What is the difference between 8.5 × 10^{-4} and 2.8 × 10^{-4}, written in standard form?

Answer:

0.00057,

Explanation:

The difference between 8.5 × 10^{-4} and 2.8 × 10^{-4 }written in standard form is as 8.5 × 10^{-4} = 0.00085

and 2.8 × 10^{-4 }= 0.00028, So 0.00085 – 0.00028 = 0.00057.

Question 14.

Mount Everest is growing at a rate of about 1.1 × 10^{-5} meter per day. Express this rate using units of a more appropriate size, and explain why the units you chose are more appropriate.

Answer:

The rate of growing is about 4 mm per year,

Explanation:

Given Mount Everest is growing at a rate of about 1.1 × 10^{-5} meter per day. Expressing this rate using

units of a more appropriate size and explaining why the units you chose are more appropriate as per given data 1 day the everest grows 1.1 × 10^{-5} meter and for 365 days it grows 365 X 1.1 X 10^{-5} = 401.5 X 10^{-5} meter = 4.015mm approximately equal to 4mm per year.

Question 15.

An elephant has a mass of 4500 kilograms. How many mice, with a mass of 2 × 10^{-2} kilogram each, would it take to equal the mass of the elephant? Write your answer in standard form.

_____________ mice

Answer:

The mass of the elephant is equal to 2,25,000 mice,

Explanation:

Given an elephant has a mass of 4500 kilograms how many mice with a mass of

2 × 10^{-2} kilogram each would it take to equal the mass of the elephant is

4,500 ÷ 2 × 10^{-2 }= 2,250 X 10^{2 }= 2,25,000 mice.^{
}