Download Today you will examine how to rewrite ex forms. You will look for

Today you will examine how to rewrite expressions with exponents in equivalent
forms. You will look for structure and patterns in the equivalent forms in order to
write them efficiently.
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1-64. In an expression like ba, b is called the base and a is called the exponent.
exponent
An exponent is shorthand for repeated multiplication. For example, the
expression n4 is equivalent to n · n · n · n.
Expand each of the expressions below. For example, to expand x3, you would
write: x · x · x.
a. y7
b. 5(2m)3
c. (x3)2
d. 4x5y2
1-65. Ms. Wang has just explained to her class how to write a simpler
expression by using the number 1. She wrote the expressions above on the board:
. Copy Ms. Wang’s steps onto your paper. Explain each step.
a. Explain why xy6 is simpler than Ms. Wang’s starting expression.
b. Rewrite each of the expressions below. Start by expanding the expressions like
Ms. Wang did, and then show how you used the Giant One to write an
equivalent expression in simpler form.
i.
ii..
iii.
iv
iv.
c. A Giant One can also be used to simplify scientific notation. Start by
expanding the expressions like Ms. Wang did, and then show how you used the
Giant One to write an equivalent expression in simpler form.
Be sure to write your simplified expression in scientific notation. See the Math
Notes box at the end of this lesson.
i.
ii.
1-66. Some of the equivalent expressions below are correct, and some are not.
Expand the expressions and use Giant Ones to justify whether or not each pair of
expressions is equivalent. If they are not equivalent, identify the mistake and correctly
simplify the expression on the left side of the equal sign.
a.
b.
c.
d.
e.
f.
g.
h.
i.
1-67. Work with your team to write four expressions with exponents, each
equivalent to x12. At least one expression must involve multiplication, one must
involve parentheses, and one must involve division. Be creative!
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Scientific Notation
Scientific notation is a way of writing very large and very small numbers compactly.
A number is said to be in scientific notation when it is written as a product of two
factors and:
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The first factor is greater than or equal to 1 but less than 10.
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The second factor is an integer power of 10.
Scientific notation usually uses the symbol “×” for multiplication instead of
using a “·” or parentheses.
For example, 2.56 × 105 is correctly written in scientific notation, but 25.6 ×
104 is not.
Scientific Notation
5.32 × 106
3.07 × 10–4
Standard Form
5,320,000
0.000307
To change 25.6 × 104 into scientific notation, first write 25.6 in scientific
notation and then simplify:
25.6
= 2.56 × 101
= 2.56 × 105
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× 104
× 104
1-68. Use what you have learned about exponents to rewrite each of the expressions
below. Homework Help ✎
a.
d. n7 · n
b. x3 · x4
e.
c. (3k5)2
f. 4xy3 · 7x2y3
1-69. Lacey and Haley are rewriting expressions in an equivalent, simpler
form. Homework Help ✎
a. Haley simplified x3 · x2 and got x5. Lacey simplified x3+ x2 and got the same result!
However, their teacher told them that only one simplification is correct. Who
simplified correctly and how do you know?
b. Haley simplifies 35 · 45 and gets the result 1210, but Lacey is not sure. Is Haley
correct? Be sure to justify your answer.
1-70. Scientific notation requires that one factor is a power of 10 and the other factor
is a number greater than or equal to 1 but less than 10. For example, 2.56 × 105 is
correctly written in scientific notation, but 25.6 × 104 is not. See the Math Notes box
in this lesson for more information. Scientific notation also uses the symbol “×” for
multiplication instead of using “·” or parentheses.
None of the numbers below is correctly written in scientific notation. Explain
why each one does not meet the criteria for scientific notation, then write it using
correct scientific notation. Homework Help ✎
. 62.5 × 103
a. 6.57 · 1000
b. 0.39 × 109
1-71. If for a certain function, f(x) = 9, find the following values. Homework
Help ✎
. f(x) + 3
a. 2f(x)
b. 2f(x) + 3
1-72. Throughout this book, key problems have been designated as
“checkpoints”. Each checkpoint problem is marked with an icon like the one at left.
These checkpoint problems are provided so that you can check to be sure you are
building skills at the expected level. When you have trouble with checkpoint
problems, refer
fer to the review materials and practice problems that are available in the
“Checkpoint Materials” section at the back of your book. Homework Help ✎
This problem is a checkpoint for solving linear equations (integer coefficients).
It will be referred to as Checkpoint 1.
Solve each equation.
a. 3x + 7 = –x – 1
b. 1 – 2x + 5 = 4x – 3
c. 4x – 2 + x = –2 + 2
d. 3x – 4 + = –2x – 5 + 5x
Check your answers by referring to the Checkpoint 1 materials located at the back
of your book.
If you needed help solving these problems correctly, then you need more
practice. Review the Checkpoint 1 materials and try the practice problems. Also
consider getting help outside of class time. From this point on, you will be expected
to do problems like this one quickly and easily.
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