Cosmic Powers Worksheet Mission: Exponentia
Cosmic Powers
Rational Exponents & Orbital Equations
Pilot:
Sector:
Module 1: Signal Conversion
PRB_01
Write in radical form: \( x^{\frac{3}{4}} \)
PRB_02
Write in exponential form: \( \sqrt[5]{y^2} \)
PRB_03
Convert & simplify: \( (16a)^{\frac{1}{2}} \)
PRB_04
Convert & simplify: \( \sqrt[3]{27x^6} \)
Module 2: Core Compression
PRB_05
Simplify: \( (x^{\frac{2}{3}})^3 \)
PRB_06
Simplify: \( (a^{-\frac{1}{2}})^4 \)
PRB_07
Evaluate: \( 16^{\frac{3}{4}} \)
PRB_08
Simplify: \( (x^{\frac{1}{2}} y^{\frac{3}{4}})^4 \)
PRB_09
Simplify: \( x^{-\frac{2}{5}} \)
PRB_10
Evaluate: \( 32^{-\frac{2}{5}} \)
PRB_11
Simplify: \( (a^{\frac{1}{4}} b^{-\frac{1}{2}})^8 \)
PRB_12
Evaluate: \( \left(\frac{8}{125}\right)^{-\frac{1}{3}} \)
Module 3: Flight Path Correction
Isolating Variables & Orbital Solutions
NAV_13
\( x^{\frac{1}{2}} = 9 \)
Calculations:
NAV_14
\( x^{\frac{3}{2}} = 27 \)
NAV_15
\( x^{-\frac{1}{3}} = \frac{1}{2} \)
NAV_16
\( (x - 5)^{\frac{1}{2}} = 4 \)
NAV_17
\( 2x^{\frac{2}{3}} = 18 \)
NAV_18
\( (3x + 1)^{\frac{1}{4}} = 2 \)
NAV_19
\( x^{\frac{3}{4}} + 5 = 13 \)
NAV_20
\( 2(x - 1)^{\frac{2}{3}} + 4 = 22 \)
Signal: Strong Firmware v4.2
MISSION EXPONENTIA READY
Cosmic Powers Answer Key Mission Control
Official Solution Manual: Cosmic Powers
Classified: Level 5 Clear
Module 1: Signal Conversion
PRB_01 \( \sqrt[4]{x^3} \)
PRB_02 \( y^{\frac{2}{5}} \)
PRB_03 \( 4\sqrt{a} \)
PRB_04 \( 3x^2 \)
Module 2: Core Compression
PRB_05 \( x^2 \)
PRB_06 \( \frac{1}{a^2} \)
PRB_07 \( 8 \)
PRB_08 \( x^2 y^3 \)
PRB_09 \( \frac{1}{x^{\frac{2}{5}}} \)
PRB_10 \( \frac{1}{4} \)
PRB_11 \( \frac{a^2}{b^4} \)
PRB_12 \( \frac{5}{2} \)
Module 3: Flight Path Correction
NAV_13 \( x = 81 \)
NAV_14 \( x = 9 \)
NAV_15 \( x = 8 \)
NAV_16 \( x = 21 \)
NAV_17 \( x = 27 \)
NAV_18 \( x = 5 \)
NAV_19 \( x = 16 \)
NAV_20 \( x = 28 \)
Flight Evaluation Briefing
Calibration Check
Verify that students have correctly switched signs for #6, #9, and #10. Negative powers must result in reciprocals before final computation.
Thruster Alignment
In the solving phase, students must raise both sides to the reciprocal exponent. Ensure they show this step to avoid gravity loss (arithmetic errors).
Endurance Scan
Problem #20 is the most complex. It requires three distinct algebraic maneuvers before the final exponent operation. Grade for process.
Unauthorized distribution of telemetry data is prohibited
Cosmic Powers Worksheet Ink Friendly Mission: Exponentia SEC_DATA_772
Cosmic Powers
Telemetry Analysis: Rational Exponents
Pilot:
Sector:
Module 1: Signal Conversion
PRB_01
Write in radical form: \( x^{\frac{3}{4}} \)
PRB_02
Write in exponential form: \( \sqrt[5]{y^2} \)
PRB_03
Convert & simplify: \( (16a)^{\frac{1}{2}} \)
PRB_04
Convert & simplify: \( \sqrt[3]{27x^6} \)
Module 2: Core Compression
PRB_05
Simplify: \( (x^{\frac{2}{3}})^3 \)
PRB_06
Simplify: \( (a^{-\frac{1}{2}})^4 \)
PRB_07
Evaluate: \( 16^{\frac{3}{4}} \)
PRB_08
Simplify: \( (x^{\frac{1}{2}} y^{\frac{3}{4}})^4 \)
PRB_09
Simplify: \( x^{-\frac{2}{5}} \)
PRB_10
Evaluate: \( 32^{-\frac{2}{5}} \)
PRB_11
Simplify: \( (a^{\frac{1}{4}} b^{-\frac{1}{2}})^8 \)
PRB_12
Evaluate: \( \left(\frac{8}{125}\right)^{-\frac{1}{3}} \)
Module 3: Flight Path Correction
Isolating Variables & Orbital Solutions
NAV_13
\( x^{\frac{1}{2}} = 9 \)
Calculations:
NAV_14
\( x^{\frac{3}{2}} = 27 \)
NAV_15
\( x^{-\frac{1}{3}} = \frac{1}{2} \)
NAV_16
\( (x - 5)^{\frac{1}{2}} = 4 \)
NAV_17
\( 2x^{\frac{2}{3}} = 18 \)
NAV_18
\( (3x + 1)^{\frac{1}{4}} = 2 \)
NAV_19
\( x^{\frac{3}{4}} + 5 = 13 \)
NAV_20
\( 2(x - 1)^{\frac{2}{3}} + 4 = 22 \)
MISSION READY FIRMWARE V4.2_ECON
Coordinates: 42.0n / 108.3w
Cosmic Powers Answer Key Ink Friendly Mission Control
Official Solution Manual: Cosmic Powers [ECON_PRINT]
Classified: Grade Ready
Module 1: Signal Conversion
PRB_01 \( \sqrt[4]{x^3} \)
PRB_02 \( y^{\frac{2}{5}} \)
PRB_03 \( 4\sqrt{a} \)
PRB_04 \( 3x^2 \)
Module 2: Core Compression
PRB_05 \( x^2 \)
PRB_06 \( \frac{1}{a^2} \)
PRB_07 \( 8 \)
PRB_08 \( x^2 y^3 \)
PRB_09 \( \frac{1}{x^{\frac{2}{5}}} \)
PRB_10 \( \frac{1}{4} \)
PRB_11 \( \frac{a^2}{b^4} \)
PRB_12 \( \frac{5}{2} \)
Module 3: Flight Path Correction
NAV_13 \( x = 81 \)
NAV_14 \( x = 9 \)
NAV_15 \( x = 8 \)
NAV_16 \( x = 21 \)
NAV_17 \( x = 27 \)
NAV_18 \( x = 5 \)
NAV_19 \( x = 16 \)
NAV_20 \( x = 28 \)
Instructor Briefing
Core Scan
Verify negative powers lead to reciprocals for #6, #9, and #10.
Path Check
Students must use reciprocal exponents to solve Modules 3.
Evaluation
#20 is the mastery problem. Award points for multi-step process.
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