Into Math Grade 8 Module 12 Review Answer Key

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: (2³)⁵ = 2¹⁵.

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 10³, 10^-3, 10¹² 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 ×10⁶ 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 • (8²)⁴. State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.
Answer:
Expression: 8⁴,

Explaantion:
Asked to write an expression with a single exponent that is equivalent to 8^-4 • (8²)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 (8²)⁴ = (8^2×4)= (8⁸), Now when two exponential terms with the same base are multiplied their powers are added while the base remains the same. So 8^-4 • (8⁸) = 8 ^-4+8 =  8⁴.

Question 6.
Compare each expression to 6⁶.

Answer:

Explanation:
Given expressions to campare with 6⁶ , So 1. (6^-1 • 6⁴)², As bases are same powers are added,
So (6^-1+4)² = (6³)², As exponent is to raise a power to a power we keep the base the same and multiply the exponents.
So (6³)² = 6^3X2 = 6⁶, So (6^-1 • 6⁴)² = 6⁶, 2.(6⁸)/(6²)² , As exponent is to raise a power to a power we
keep the base the same and multiply the exponents. So (6⁸)/(6^2X2)  = (6⁸)/(6⁴) , As to divide two powers with the same base, keep the base the same and subtract the exponents.
So (6^{8 – 4}) = 6⁴ ,So as 6⁴ < 6⁶, 3. (6²/6⁰)⁴ , As to divide two powers with the same base,
keep the base the same and subtract the exponents.
So (6^2-0)⁴ ,= (6²)⁴ , Now when exponent is to raise a power to a power we keep the base the same and
multiply the exponents so (6²)⁴  = 6^2x4  = 6⁸ > 6⁶, Completed teh table as shown above.

Question 7.
Select all the expressions equivalent to \(\frac{9^{3} \cdot 9^{5}}{9^{2}}\).
(A) 3⁸
(B) 3¹²
(C) 9⁶
(D) 9⁴
(E) 27²
(F) 27⁴
Answer:
(B) 3¹², (C) 9⁶ and (F) 27⁴,

Explanation:
Given \(\frac{9^{3} \cdot 9^{5}}{9^{2}}\) when bases are same powers are added so (9^{3 + 5}) / 9² = 9⁸/ 9² , now as to divide two powers with the same base keep the base the same and subtract the exponents. So 9⁸/ 9² = 9^8-2 = 9⁶, 9⁶ is equivalent to (3 X 3)⁶ = (3¹ X 3¹)⁶ = (3¹⁺¹)⁶ = 3^2X6 = 3¹² and 9⁶ 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) 3¹², (C) 9⁶ and (F) 27⁴.

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 4¹= 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 × 10⁴ 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 × 10⁴ 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 × 10⁴ 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 × 10ⁿ = 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 × 10⁵ is greater than 3.2 × 10⁶. Is Naomi correct? Explain your reasoning.
Answer:t
No, Naomi is incorrect,

Explanation:
Given to find Naomi says that 9.8 × 10⁵ is greater than 3.2 × 10⁶.
Is Naomi correct, No Naomi is incorrect as 9.8 × 10⁵ is not greater than 3.2 × 10⁶. because if we see 3.2 X 10⁶ = 32 × 10⁵, now if compare both sides we have 10⁵ and now if we compare
9.8 and 32, 32 is greater than 9.8, So 3.2 × 10⁶ is greater than 9.8 × 10⁵, 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 × 10¹
(B) 4 × 10⁴
(C) 4 × 10⁵
(D) 4 × 10⁸
Answer:
(D) 4 × 10⁸,

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 10³⁺³/3 X 10^-2 =
12 X 10⁶/3 X 10^-2 to divide two powers with the same base, keep the base the same and subtract the exponents so 4 X 10⁶⁺² = 4 X 10⁸ 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 10² = 2,25,000 mice.