During a deep-space mission, Emek and Tara noticed that messages from their research probe were arriving with gaps. The diagnostic system modeled the weakened signal as (3^2)^3 times 3^-4 signal units. Tara first used the power-of-a-power rule to rewrite (3^2)^3 as 3^6. Emek then combined powers with the same base: 3^6 times 3^-4 = 3^2, so the received signal level was 9 units. Reliable communication required at least 27 units. The amplifier offered settings that multiplied the signal by 3, 6, or 9. Emek calculated that the lowest setting would raise the signal to 9 times 3 = 27 units, exactly meeting the requirement. Tara checked the original expression another way, replacing 3^-4 with 1/81, and again obtained 9 units before amplification. Confident in their calculation, they selected the factor-of-3 setting. The next transmission arrived clearly, restoring reliable communication without using unnecessary power.

Challenges

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Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Recall the rule for raising a power to another power.
A student simplifies the probe's signal expression (3^2)^3 times 3^-4 as 3^(2 + 3 - 4) = 3. Which statement identifies the student's error?
The student should change 3^-4 to 3^4 before combining the powers.The student should multiply 2 by 3 when simplifying (3^2)^3, giving 3^6 before combining the powers.The student should multiply all three exponents, giving 3^(2 times 3 times -4).The student should add 4 instead of subtracting 4 because the exponent is negative.
Correct. For a power raised to a power, the exponents are multiplied.
Not quite. Check how the exponent 3 affects the exponent 2 in (3^2)^3.
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Hint
First use the power-of-a-power rule, then combine powers with the same base.
During a later transmission, the diagnostic system models the signal as (3^2)^4 times 3^-5 signal units. What is the signal level in units?
Correct. The later signal level is 27 units.
Check the power-of-a-power rule first, then add the exponents when multiplying powers with the same base.
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Hint
Evaluate (3^2)^3 and use the fact that a negative exponent represents a reciprocal.
Tara checks the original expression by replacing 3^-4 with 1/81. Which calculation correctly verifies that the original signal level was 9 units?
(3^2)^3 times 81 = 729 times 81(3^2)^3 times 1/81 = 81/729(3^2)^3 times 1/81 = 729/81 = 9(3^2)^3 times 1/81 = 243/81 = 3
Correct. The alternative calculation gives 729/81 = 9, confirming the original result.
Not quite. Evaluate (3^2)^3 carefully and then multiply by 1/81.
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Derin Uzay Sinyalini Onarmak
TR
ARC_G09_OPS_02_STORY_03
Derin Uzay Sinyalini Onarmak
Repairing the Deep-Space Signal
EN
ARC_G09_OPS_02_STORY_03
Repairing the Deep-Space Signal