Answer :
We need to find the exponent for the power of 10 that makes the equation in each part true. Let’s solve each one step by step.
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Part (a):
We have
[tex]$$2.77 \times 10^x = 277000.$$[/tex]
Isolate [tex]$10^x$[/tex] by dividing both sides by 2.77:
[tex]$$10^x = \frac{277000}{2.77}.$$[/tex]
Calculating the quotient gives:
[tex]$$\frac{277000}{2.77} = 100000.$$[/tex]
Since
[tex]$$10^5 = 100000,$$[/tex]
we have
[tex]$$x = 5.$$[/tex]
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Part (b):
The equation is
[tex]$$7.4 \div 10^x = 0.00074.$$[/tex]
Write this as:
[tex]$$\frac{7.4}{10^x} = 0.00074.$$[/tex]
Multiplying both sides by [tex]$10^x$[/tex], we obtain:
[tex]$$7.4 = 0.00074 \times 10^x.$$[/tex]
Now, solve for [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{7.4}{0.00074}.$$[/tex]
Calculating the fraction:
[tex]$$\frac{7.4}{0.00074} = 10000.$$[/tex]
Since
[tex]$$10^4 = 10000,$$[/tex]
we deduce that
[tex]$$x = 4.$$[/tex]
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Part (c):
The equation is
[tex]$$9 \times 10^x = 9000000000.$$[/tex]
Divide both sides by 9 to isolate [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{9000000000}{9}.$$[/tex]
This gives
[tex]$$10^x = 1000000000.$$[/tex]
Since
[tex]$$10^9 = 1000000000,$$[/tex]
we have
[tex]$$x = 9.$$[/tex]
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Part (d):
We start with
[tex]$$2.48 \div 10^x = 0.0000248.$$[/tex]
Rewrite it as:
[tex]$$\frac{2.48}{10^x} = 0.0000248.$$[/tex]
Multiply both sides by [tex]$10^x$[/tex]:
[tex]$$2.48 = 0.0000248 \times 10^x.$$[/tex]
Now, solve for [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{2.48}{0.0000248}.$$[/tex]
Performing the calculation yields:
[tex]$$10^x = 100000.$$[/tex]
Since
[tex]$$10^5 = 100000,$$[/tex]
it follows that
[tex]$$x = 5.$$[/tex]
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Part (e):
The equation is
[tex]$$9.1 \times 10^x = 910.$$[/tex]
Divide both sides by 9.1:
[tex]$$10^x = \frac{910}{9.1}.$$[/tex]
This simplifies to:
[tex]$$10^x = 100.$$[/tex]
Since
[tex]$$10^2 = 100,$$[/tex]
we deduce that
[tex]$$x = 2.$$[/tex]
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Part (f):
We have the equation
[tex]$$39.4 \div 10^x = 0.0394.$$[/tex]
Express it as:
[tex]$$\frac{39.4}{10^x} = 0.0394.$$[/tex]
Multiply both sides by [tex]$10^x$[/tex]:
[tex]$$39.4 = 0.0394 \times 10^x.$$[/tex]
Now, solve for [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{39.4}{0.0394}.$$[/tex]
This gives:
[tex]$$10^x = 1000.$$[/tex]
And since
[tex]$$10^3 = 1000,$$[/tex]
we find
[tex]$$x = 3.$$[/tex]
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Final Answers:
a) [tex]$x = 5$[/tex]
b) [tex]$x = 4$[/tex]
c) [tex]$x = 9$[/tex]
d) [tex]$x = 5$[/tex]
e) [tex]$x = 2$[/tex]
f) [tex]$x = 3$[/tex]
--------------------------------------------
Part (a):
We have
[tex]$$2.77 \times 10^x = 277000.$$[/tex]
Isolate [tex]$10^x$[/tex] by dividing both sides by 2.77:
[tex]$$10^x = \frac{277000}{2.77}.$$[/tex]
Calculating the quotient gives:
[tex]$$\frac{277000}{2.77} = 100000.$$[/tex]
Since
[tex]$$10^5 = 100000,$$[/tex]
we have
[tex]$$x = 5.$$[/tex]
--------------------------------------------
Part (b):
The equation is
[tex]$$7.4 \div 10^x = 0.00074.$$[/tex]
Write this as:
[tex]$$\frac{7.4}{10^x} = 0.00074.$$[/tex]
Multiplying both sides by [tex]$10^x$[/tex], we obtain:
[tex]$$7.4 = 0.00074 \times 10^x.$$[/tex]
Now, solve for [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{7.4}{0.00074}.$$[/tex]
Calculating the fraction:
[tex]$$\frac{7.4}{0.00074} = 10000.$$[/tex]
Since
[tex]$$10^4 = 10000,$$[/tex]
we deduce that
[tex]$$x = 4.$$[/tex]
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Part (c):
The equation is
[tex]$$9 \times 10^x = 9000000000.$$[/tex]
Divide both sides by 9 to isolate [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{9000000000}{9}.$$[/tex]
This gives
[tex]$$10^x = 1000000000.$$[/tex]
Since
[tex]$$10^9 = 1000000000,$$[/tex]
we have
[tex]$$x = 9.$$[/tex]
--------------------------------------------
Part (d):
We start with
[tex]$$2.48 \div 10^x = 0.0000248.$$[/tex]
Rewrite it as:
[tex]$$\frac{2.48}{10^x} = 0.0000248.$$[/tex]
Multiply both sides by [tex]$10^x$[/tex]:
[tex]$$2.48 = 0.0000248 \times 10^x.$$[/tex]
Now, solve for [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{2.48}{0.0000248}.$$[/tex]
Performing the calculation yields:
[tex]$$10^x = 100000.$$[/tex]
Since
[tex]$$10^5 = 100000,$$[/tex]
it follows that
[tex]$$x = 5.$$[/tex]
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Part (e):
The equation is
[tex]$$9.1 \times 10^x = 910.$$[/tex]
Divide both sides by 9.1:
[tex]$$10^x = \frac{910}{9.1}.$$[/tex]
This simplifies to:
[tex]$$10^x = 100.$$[/tex]
Since
[tex]$$10^2 = 100,$$[/tex]
we deduce that
[tex]$$x = 2.$$[/tex]
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Part (f):
We have the equation
[tex]$$39.4 \div 10^x = 0.0394.$$[/tex]
Express it as:
[tex]$$\frac{39.4}{10^x} = 0.0394.$$[/tex]
Multiply both sides by [tex]$10^x$[/tex]:
[tex]$$39.4 = 0.0394 \times 10^x.$$[/tex]
Now, solve for [tex]$10^x$[/tex]:
[tex]$$10^x = \frac{39.4}{0.0394}.$$[/tex]
This gives:
[tex]$$10^x = 1000.$$[/tex]
And since
[tex]$$10^3 = 1000,$$[/tex]
we find
[tex]$$x = 3.$$[/tex]
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Final Answers:
a) [tex]$x = 5$[/tex]
b) [tex]$x = 4$[/tex]
c) [tex]$x = 9$[/tex]
d) [tex]$x = 5$[/tex]
e) [tex]$x = 2$[/tex]
f) [tex]$x = 3$[/tex]